A HISTORY
OF
GREEK MATHEMATICS
*
VOLUME I
Digitized by the Internet Archive
in 2011 with funding from
University of Ottawa
http://www.archive.org/details/historyofgreekm01heat
A HISTORY
OF
GREEK MATHEMATICS
BY
SIR THOMAS HEATH
K.C.B., K.C.V.O., F.R.S.
SC.D. CAME. ; HON. D.SC. OXFORD
HONORARY FELLOW (FORMRRLV FELLOW) OF TRIVTTV COLLEflE, CAMBRIDGE
• ... An independent world,
Created out of pure intelligence/
Wordsworth.
VOLUME I
FROM THALES TO EUCLID
OXFORD
AT THE CLARENDON PRESS
1921
'JAN 9 1951
OXFORD UNIVERSITY PRESS
London Edinburgh Glasgow Copenhagen
New York Toronto Melbourne Cape Town
Bombay Calcutta Madras Shanghai
HUMPHREY MILFORD
Publisher to the University
16?S
PREFACE
The idea may seem quixotic, but it is nevertheless the
author's confident hope that this book will give a fresh interest
to the story of Greek mathematics in the eyes both of
mathematicians and of classical scholars.
For the mathematician the important consideration is that
the foundations of mathematics and a great portion of its
content are Greek. The Greeks laid down the first principles,
invented the" methods ab initio, and fixed the terminology.
Mathematics in short is a Greek science, whatever new
developments modern analysis has brought or may bring.
The interest of the subject for the classical scholar is no
doubt of a different kind. Greek mathematics reveals an
important aspect of the Greek genius of which the student of
Greek culture is apt to lose sight. Most people, when they
think of the Greek genius, naturally call to mind its master-
pieces in literature and art with their notes of beauty, truth,
freedom and humanism. But the Greek, with his insatiable
desire to know the true meaning of everything in the uni-
verse and to be able to give a rational explanation of it, was
just as irresistibly driven to natural science, mathematics, and
exact reasoning in general or logic. This austere side of the
Greek genius found perhaps its most complete expression in
Aristotle. Aristotle would, however, by no means admit that
mathematics was divorced from aesthetic ; he could conceive,
he said, of nothing more beautiful than the objects of mathe-
matics. Plato delighted in geometry and in the wonders of
numbers ; dyecofjiiTprjTos nySels elcriT<o, said the inscription
over the door of the Academy. Euclid was a no less typical
Greek. Indeed, seeing that so much of Greek is mathematics,
vi PREFACE
it is arguable that, if one would understand the Greek genius
fully, it would be a good plan to begin with their geometry.
The story of , Greek mathematics has been written before.
Dr. James Gow did a great service by the publication in 1884
of his Short History of Greek Mathematics, a scholarly and
useful work which has held its own and has been quoted with
respect and appreciation by authorities on the history of
mathematics in all parts of the world. At the date when he
wrote, however, Dr. Gow had necessarily to rely upon the
works of the pioneers Bretschneider, Hankel, Allman, and
Moritz Cantor (first edition). Since then the subject has been
very greatly advanced ; new texts have been published, im-
portant new documents have been discovered, and researches
by scholars and mathematicians in different countries have
thrown light on many obscure points. It is, therefore, high
time for the complete story to be rewritten.
It is true that in recent years a number of attractive
histories of mathematics have been published in England and
America, but these have only dealt with Greek mathematics
as part of the larger subject, and in consequence the writers
have been precluded, by considerations of space alone, from
presenting the work of the Greeks in sufficient detail.
The same remark applies to the German histories of mathe-
matics, even to the great work of Moritz Cantor, who treats
of the history of Greek mathematics in about 400 pages of
vol. i. While no one would wish to disparage so great a
monument of indefatigable research, it was inevitable that
a book on such a scale would in time prove to be inadequate,
and to need correction in details ; and the later editions have
unfortunately failed to take sufficient account of the new
materials which have become available since the first edition
saw the light.
The best history of Greek mathematics which exists at
present is undoubtedly that of Gino Loria under the title
Le scietize esatte nelV anticct Grecla (second edition 1914,
PREFACE vii
Ulrico Hocpli, Milano). Professor Loria arranges his material
in five Books, (1) on pre-Euclidean geometry, (2) on the
Golden Age of Greek geometry (Euclid to Apollonius), (3) on
applied mathematics, including astronomy, sphaeric, optics,
&c, (4) on the Silver Age of Greek geometry, (5) on the
arithmetic of the Greeks. Within the separate Books the
arrangement is chronological, under the names of persons or
schools. I mention these details because they raise the
question whether, in a history of this kind, it is best to follow
chronological order or to arrange the material according to
subjects, and, if the latter, in what sense of the word 'subject'
and within what limits. As Professor Loria says, his arrange-
ment is ' a compromise between arrangement according to
subjects and a strict adherence to chronological order, each of
which plans has advantages and disadvantages of its own '.
In this book I have adopted a new arrangement, mainly
according to subjects, the nature of which and the reasons for
which will be made clear by an illustration. Take the case of
a famous problem which plays a great part in the history of
Greek geometry, the doubling of the cube, or its equivalent,
the finding of two mean proportionals in continued proportion
between two given straight lines. Under a chronological
arrangement this problem comes up afresh on the occasion of
each new solution. Now it is obvious that, if all the recorded
solutions are collected together, it is much easier to see the
relations, amounting in some cases to substantial identity,
between them, and to get a comprehensive view of the history
of the problem. I have therefore dealt with this problem in
a separate section of the chapter devoted to ' Special Problems',
and I have followed the same course with the other famous
problems of squaring the circle and trisecting any angle.
Similar considerations arise with regard to certain well-
defined subjects such as conic sections. It would be incon-
venient to interrupt the account of Menaechmus's solution
of the problem of the two mean proportionals in order to
ft
viii PREFACE
consider the way in which he may have discovered the conic
sections and their fundamental properties. It seems to me
much better to give the complete story of the origin and
development of the geometry of the conic sections in one
place, and this has been done in the chapter on conic sections
associated with the name of Apollonius of Perga. Similarly
a chapter has been devoted to algebra (in connexion with
Diophantus) and another to trigonometry (under Hipparchus,
Menelaus and Ptolemy).
At the same time the outstanding personalities of Euclid
and Archimedes demand chapters to themselves. Euclid, the
author of the incomparable Elements, wrote on almost all
the other branches of mathematics known in his day. Archi-
medes's work, all original and set forth in treatises which are
models of scientific exposition, perfect in form and style, was
even wider in its range of subjects. The imperishable and
unique monuments of the, genius of these two men must be
detached from their surroundings and seen as a whole if we
would appreciate to the full the pre-eminent place which they
occupy, and will hold for all time, in the history of science.
The arrangement which I have adopted necessitates (as does
any other order of exposition) a certain amount of repetition
and cross-references ; but only in this way can the necessary
unity be given to the whole narrative.
One other point should be mentioned. It is a defect in the
existing histories that, while they state generally the contents
of, and the main propositions proved in, the great treatises of
Archimedes and Apollonius, they make little attempt to
describe the procedure by which the results are obtained.
I have therefore taken pains, in the most significant cases,
to show the course of the argument in sufficient detail to
enable a competent mathematician to grasp the method used
and to apply it, if he will, to other similar investigations.
The work was begun in 1913, but the bulk of it was
written, as a distraction, during the first three years of the
PREFACE
IX
war, the hideous course of which seemed day by day to
enforce the profound truth conveyed in the answer of Plato
to the Delians. When they consulted him on the problem set
them by the Oracle, namely that of duplicating the cube, he
replied, ; It must be supposed, not that the god specially
wished this problem solved, but that he would have the
Greeks desist from war and wickedness and cultivate the
Muses, so that, their passions being assuaged by philosophy
and mathematics, they might live in innocent and mutually
helpful intercourse with one another '.
Truly
Greece and her foundations are
Built below the tide of war,
Based on the crystalline sea
Of thought and its eternity.
T. L. H.
CONTENTS OF VOL. I
I. INTRODUCTORY
PAGES 1-25
The Greeks and mathematics ......
Conditions favouring development of philosophy among the
Greeks
Meaning and classification of mathematics
(a) Arithmetic and logistic ....
(3) Geometry and geodaesia ....
(y) Physical subjects, mechanics, oplics, &c. .
Mathematics in Greek education
1-3
3-10
10-18
13-16
16
17-18
18-25
II. GREEK NUMERICAL NOTATION AND ARITHMETICAL
OPERATIONS 26-64
The decimal system 26-27
Egyptian numerical notation 27-28
Babylonian systems
(a) Decimal. (/3) Sexagesimal ..... 28-29
Greek numerical notation 29-45
(a) The 'Herodianic' signs 30-31
(/3) The ordinary alphabetic numerals .... 31-35
(y) Mode of writing numbers in the ordinary alphabetic
notation ........ 36-37
(6 V ) Comparison of the two systems of numerical notation 37-39
(e) Notation for large numbers ..... 39-41
(i) Apollonius's 'tetrads' 40
(ii) Archimedes's system (by octads) . . . 40-41
Fractions
(a) The Egyptian system 41-42
(/3) The ordinary Greek form, variously written . . 42-44
(y) Sexagesimal fractions 41-45
Practical calculation
(a) The abacus 46-52
(/3) Addition and subtraction 52
(*y) Multiplication
(i) The Egyptian method 52-53
(ii) The Greek method . . . . . 53-54
(iii) Apollonius's continued multiplications . . 54-57
(iv) Examples of ordinary multiplications . . 57-58
(8) Division . ' . . . * 58-60
(f) Extraction of the square root ..... 60-63
(£) Extraction of the cube root . . . . . 63-64
XI 1
CONTENTS
PAGES 65 117
67-69
69-70
70-74
74-76
III. PYTHAGOREAN ARITHMETIC
Numbers and the universe
Definitions of the unit and of number
Classification of numbers .
' Perfect ' and ' Friendly ' numbers .
Figured numbers
(a) Triangular numbers 76-77
(8) Square numbers and gnomons. .... 77
(y) History of the term 'gnomon' .... 78-79
(8) Gnomons of the polygonal numbers ... 79
(f) Right-angled triangles with sides in rational
numbers 79-82
(£) Oblong numbers 82-84.
The theory of proportion and means . . ... 84-90
(a) Arithmetic, geometric and harmonic means . . 85-86
(8) Seven other means distinguished .... 86-89
(y) Plato on geometric means between two squares or
two cubes .89-90
(ft) A theorem of Archytas 90
The k irrational ' 90-91
Algebraic equations
(a) ' Side- ' and ' diameter- ' numbers, giving successive
approximations to «/2 (solutions of 2 x 1 — }/— ± 1) 91-93
(8) The inavBrHJui (' bloom') of Thymaridas . . "' . 94-96
(y) Area of rectangles in relation to perimeter (equation
xy = 2x + y) 96-97
Systematic treatises on arithmetic (theory of numbers) . 97-115
Nicomachus. Introdnctio Arithmetica .... 97-112
Sum of series of cube numbers .... 108-110
Theon of Smyrna 112-118
Iamblichus, Commentary on Nicomachus . . . 113-115
The pythmen and the rule of nine or seven . . . 115-117
IV. THE EARLIEST GREEK GEOMETRY. THALES . . 118-140
The ' Summary ' of Proclus 118-121
Tradition as to the origin of geometry .... 121-122
Egyptian geometry, i.e. mensuration .... 122-128
The beginnings of Greek geometry. Thales . . . 128-139
(a) Measurement of height of pyramid . . . . 129-130
(8) Geometrical theorems attributed to Thales . . 130-137
(y) Thales as astronomer 137-139
From Thales to Pythagoras 139-140
V. PYTHAGOREAN GEOMETRY ....
Pythagoras .......
Discoveries attributed to the Pythagoreans
(a) Equality of sum of angles of any triangle
right angles .....
(/3) The ' Theorem of Pythagoras ' . .
(y) Application of areas and geometrical algebr
tion of quadratic equations)
(8) The irrational ......
(e) The five regular solids ....
(£) Pythagorean astronomy ....
Recapitulation .......
to two
a (solu-
141 169
141-142
143-144
144-149
150-154
154-157
158-162
162-165
165-169
CONTENTS
xm
VI.
PROGRESS IN THE ELEMENTS DOWN TO PLATO'S
TIME PAGES
Extract from Proclus's summary .
Anaxagoras ......
Oenopides of Chios .....
Democritus ......
Hippias of Elis
Hippocrates of Chios ....
(a) Hippocrates's quadrature of lunes .
(/3) Reduction of the problem of doubling the cube to
the finding of two mean proportionals
(y) The Elements as known to Hippocrates
Theodoras of Cyrene ....
Theaetetus . . . . .
Archytas .......
Summary .......
VII. SPECIAL PROBLEMS ....
The squaring of the circle
Antiphon
Bryson .......
Hippias, Dinostratus, Nicomedes. &c. .
(a) The quadratrix of Hippias .
O) The spiral of Archimedes .
(y) Solutions by Apollonius and Carpus
(3) Approximations to the value of it
The trisection of any angle
(a) Reduction to a certain mxxris, solved by conies
(3) The vevais equivalent to a cubic equation
(y) The conchoids of Nicomedes ....
(ft) Another reduction to a vevais (Archimedes) .
(e) Direct solutions by means of conies (Pappus)
The duplication of the cube, or the problem of the two
mean proportionals
(a) History of the problem ....
(/3) Archytas ......
(y) Eudoxus
(h) Menaechmus ......
(f) The solution attributed to Plato .
(() Eratosthenes
(/;) Nicomedes ......
(6) Apollonius, Heron, Philon of Byzantium
(i) Diocles and the cissoid ....
(k) Sporus and Pappus ....
(X) Approximation to a solution by plane methods only
170-217
170-172
172-174
174-176
176-181
182
182-202
183-200
200-201
201-202
202-209
209-212
213-216
216-217
218-270
220-235
221-223
223-225
225-226
226-230
230-231
231-232
232-235
235-244
235-237
237-238
238-240
240-241
241-244
244-270
244-246
246-249
249-251
251-255
255-258
258-260
260-262
262-264
264-266
266-268
268-270
VIII. ZENO OF ELEA .
Zeno's arguments about motion
IX. PLATO
Contributions to the philosophy of mathematics
(a) The hypotheses of mathematics
(/3) The two intellectual methods
(y) Definitions
271-283
273-283
284-315
288-294
289-290
290-292
292-294
xiv CONTENTS
IX. Continued
Summary of the mathematics in Plato . . pages 294-308
(a) Regular and semi-regular solids .... 294-295
(/3) The construction of the regular solids . . . 296-297
(y) Geometric means between two square numbers or
two cubes 297
(d) The two geometrical passages in the Meno . . 297-303
(e) Plato and the doubling of the cube . . . 303
(() Solution of x 2 + y 2 = z 1 in integers . . . 304
(v) Incommensurables 304-305
(0) The Geometrical Number 305-308
Mathematical ' arts ' 308-315
(a) Optics 309
(0) Music 310
(>) Astronomy 310-315
X. FROM PLATO TO EUCLID 316-353
Heraclides of Pontus : astronomical discoveries . . 316-317
Theory of numbers (Speusippus, Xenocrates) . . . 318-319
The Elements. Proclus's summary (continued) . . 319-321
Eudoxus 322-335
(a) Theory of proportion 325-327
(£) The method of exhaustion 327-329
(y) Theory of concentric spheres .... 329-335
Aristotle 335-348
(«) First principles 336-338
(3) Indications of proofs differing from Euclid's. . 338-340
(■)) Propositions not found in Euclid .... 340-341
(h) Curves and solids known to Aristotle . . . 341-342
(e) The continuous and the infinite .... 342-344
(0 Mechanics 344-346
The Aristotelian tract on indivisible lines . . . 346-348
Sphaeric
Autolycus of Pitane 348-353
A lost text-book on Sphaeric 349-350
Autolycus, On the Moving Sphere: relation to Euclid . 351-352
Autolycus, On Risings and Settings .... 352-353
XI. EUCLID 354-446
Date and traditions 354-357
Ancient commentaries, criticisms and references . . 357-360
The text of the Elements 360-361
Latin and Arabic translations 361-364
The first printed editions 364-365
The study of Euclid in the Middle Ages .... 365-369
The first'English editions 369-370
Technical terms
(a) Terms for the formal divisions of a proposition . 370-371
(j3) The diopLOTfxos or statement of conditions of possi-
bility . . . 371
(y) Analysis, synthesis, reduction, reductio ad absurdum 371-372
(d) Case, objection, porism, lemmn 372-373
Analysis of the Elements
Book I 373-379
..II 379-380
CONTENTS
XV
Book III
pages 380-383
„ IV
. 383-384
„ v
. 384-391
„ VI
. 391-397
„ VII
. 397-399
„ VIII
. 399-400
„ IX
. 400-402
„ x
. 402-412
„ XI
. 412-413
„ XII
. 413-415
„ XIII ....
. 415-419
The so-called Books XIV, XV .
. 419-421
The Data
. 421-425
On divisions {of figures) .
. 425-430
Lost geometrical works
(a) The Pseudaria .
. 430-431
(/3) The Porisms ....
. 431-438
(y) The Conies ....
. 438-439
(ft) The Surface Loci .
Applied mathematics
. 439-440
(«) The Phaenomena
. 440-441
0) Optics and Catoptrica
(y) Music
. 441-444
. 444-445
(5) Works on mechanics attributed to
Euclid . . 445-446
ERRATA
Vol. i, p. 120, line 7, for i Laodamas ' read ' Leodamas '.
Vol. i, p 161, line 5 from foot, for l pentagon ' read 4 pentagram \
Vol. i, p. 290, line 9 from foot, for * ideals' read 'ideas'.
Vol. ii, p. 324, note 2, line 12, for ' 1853' read ' 1851 '.
Vol. ii, p. 360, line 8 from foot, for ' Breton le Champ' read ' Breton
de Champ'.
INTRODUCTORY
The Greeks and mathematics.
It is an encouraging s igm of the times that more and more
effort is being directed to promoting a due appreciation and
a clear understanding of the gifts of the Greeks to mankind.
What we owe to Greece, what the Greeks have done for
civilization, aspects of the Greek genius : such are the themes
of many careful studies which have made a wide appeal and
will surely produce their effect. In truth all nations, in the
West at all events, have been to school to the Greeks, in art,
literature, philosophy, and science, the things which are essen-
tial to the rational use and enjoyment of human powers and
activities, the things which make life worth living to a rational
human being. \ Of all peoples the Greeks have dreamed the
dream of life the best.' And the Greeks were not merely the
pioneers in the branches of knowledge which they invented
and to which they gave names. What they began they carried
to a height of perfection which has not since been surpassed ;
if there are exceptions, it is only where a few crowded centuries
were not enough to provide the accumulation of experience
required, whether for the purpose of correcting hypotheses
which at first could only be of the nature of guesswork, or of
suggesting new methods and machinery.
Of all the manifestations of the Greek genius none is more
impressive and even awe-inspiring than that which is revealed
by the history of Greek mathematics. Not only are the range
and the sum of what the Greek mathematicians actually
accomplished wonderful in themselves ; it is necessary to bear
in mind that this mass of original work was done in an almost
incredibly short space of time, and in spite of the comparative
inadequacy (as it would seem to us) of the only methods at
their disposal, namely those of pure geometry, supplemented,
where necessary, by the ordinary arithmetical operations.
1523 B
2 INTRODUCTORY
Let us, confining ourselves to the main subject of pure
geometry by way of example, anticipate so far as to mark
certain definite stages in its development, with the intervals
separating them. In Thales's time (about 600 B. c.) we find
the first glimmerings of a theory of geometry, in the theorems
that a circle is bisected by any diameter, that an isosceles
triangle has the angles opposite to the equal sides equal, and
(if Thales really discovered this) that the angle in a semicircle
is a right angle. Rather more than half a century later
Pythagoras was taking the first steps towards the theory of
numbers and continuing the work of making geometry a
theoretical science ; he it was who first made geometry one of
the subjects of a liberal education. The Pythagoreans, before
the next century was out (i. e. before, say, 450 B. c), had practi-
cally completed the subject-matter of Books I -II, IV, VI (and
perhaps III) of Euclid's Elements, including all the essentials
of the * geometrical algebra ' which remained fundamental in
Greek geometry ; the only drawback was that their theory of
proportion was not applicable to incommensurable but only
to commensurable magnitudes, so that it proved inadequate
as soon as the incommensurable came to be discovered.
In the same fifth century the difficult problems of doubling
the cube and trisecting any angle, which are beyond the
geometry of the straight line and circle, were not only mooted
but solved theoretically, the former problem having been first
reduced to that of finding two mean proportionals in continued
proportion (Hippocrates of Chios) and then solved by a
remarkable construction in three dimensions (Archytas), while
the latter was solved by means of the curve of Hippias of
Elis known as the quadratrix ; the problem of squaring the
circle was also attempted, and Hippocrates, as a contribution
to it, discovered and squared three out of the five lunes which
can be squared by means of the straight line and circle. In
the fourth century Eudoxus discovered the great theory of
proportion expounded in Euclid, Book V, and laid down the
principles of the method of exhaustion for measuring areas and
volumes ; the conic sections and their fundamental properties
were discovered by Menaechmus; the theory of irrationals
(probably discovered, so far as V2 is concerned, by the
early Pythagoreans) was generalized by Theaetetus ; and the
THE GREEKS AND MATHEMATICS 3
geometry of the sphere was worked out in systematic trea-
tises. About the end of the century Euclid wrote his
Elements in thirteen Books. The next century, the third,
is that of Archimedes, who may be said to have anticipated
the integral calculus, since, by performing what are practi-
cally integrations, he found the area of a parabolic segment
and of a spiral, the surface and volume of a sphere and a
segment of a sphere, the volume of any segment of the solids
of revolution of the second degree, the centres of gravity of
a semicircle, a parabolic segment, any segment of a paraboloid
of revolution," and any segment of a sphere or spheroid.
Apollonius of Perga, the ' great geometer ', about 200 B. c,
completed the theory of geometrical conies, with specialized
investigations of normals as maxima and minima leading
quite easily to the determination of the circle of curvature
at any point of a conic and of the equation of the evolute of
the conic, which with us is part of analytical conies. With
Apollonius the main body of Greek geometry is complete, and
we may therefore fairly say that four centuries sufficed to
complete it.
But some one will say, how did all this come about? What
special aptitude had the Greeks for mathematics ? The answer
to this question is that their genius for mathematics was
simply one aspect of their genius for philosophy. Their
mathematics indeed constituted a large part of their philo-
sophy down to Plato. Both had the same origin.
Conditions favouring the development of philosophy
among the Greeks.
All men by nature desire to know, says Aristotle. 1 The
Greeks, beyond any other people of antiquity, possessed the
love of knowledge for its own sake ; with them it amounted
to an instinct and a passion. 2 We see this first of all in their
love of adventure. It is characteristic that in the Odyssey
Odysseus is extolled as the Ijero who had ' seen the cities of
many men and learned their mind', 3 often even taking his life
in his hand, out of a pure passion for extending his horizon
1 Arist. Metaph. A. 1, 980 a 21.
2 Cf. Butcher, Some Aspects of the Greek Genius, 1892, p. J,
3 Od. i. 3.
B 2
4 INTRODUCTORY
as when he went to see the Cyclopes in order to ascertain ' what
sort of people they were, whether violent and savage, with no
sense of justice, or hospitable and godfearing'. 1 Coming
nearer to historical times, we find philosophers and statesmen
travelling in order to benefit by all the wisdom that other
nations with a longer history had gathered during the cen-
turies. Thales travelled in Egypt and spent his time with
the priests. Solon, according to Herodotus, 2 travelled ' to see
the world' (Oeoopirjs eivtKev), going to Egypt to the court of
Amasis, and visiting Croesus at Sardis. At Sardis it was not
till ' after he had seen and examined everything ' that he had
the famous conversation with Croesus ; and Croesus addressed
him as the Athenian of whose wisdom and peregrinations he
had heard great accounts, proving that he had covered much
ground in seeing the world and pursuing philosophy.
(Herodotus, also a great traveller, is himself an instance of
the capacity of the Greeks for assimilating anything that
could be learnt from any other nations whatever; and,
although in Herodotus's case the object in view was less the
pursuit of philosophy than the collection of interesting infor-
mation, yet he exhibits in no less degree the Greek passion
for seeing things as they are and discerning their meaning
and mutual relations ; ' he compares his reports, he weighs the
evidence, he is conscious of his own office as an inquirer after
truth'.) But the same avidity for learning is best of all
illustrated by the similar tradition with regard to Pythagoras's
travels. Iamblichus, in his account of the life of Pythagoras, 3
says that Thales, admiring his remarkable ability, communi-
cated to him all that he knew, but. pleading his own age and
failing strength, advised him for his better instruction to go
and study with the Egyptian priests. Pythagoras, visiting
Sidon on the way, both because it was his birthplace and
because he properly thought that the passage to Egypt would
be easier by that route, consorted there with the descendants
of Mochus, the natural philosopher and prophet, and with the
other Phoenician hierophants, and was initiated into all
the rites practised in Biblus, Tyre, and in many parts of
Syria, a regimen to which he submitted, not out of religious
1 Od. ix. 174-6. 2 Herodotus, i. 30.
3 Iamblichus, De vita Pythagorica, cc. 2-4.
DEVELOPMENT OF PHILOSOPHY 5
enthusiasm, 'as you might think' (a>? av ns ocrrXco? i>7roAa/?oi),
but much more through love and desire for philosophic
inquiry, and in order to secure that he should not overlook
any fragment of knowledge worth acquiring that might lie
hidden in the mysteries or ceremonies of divine worship ;
then, understanding that what he found in Phoenicia was in
some sort an offshoot or descendant of the wisdom of the
priests of Egypt, he concluded that he should acquire learning
more pure and more sublime by going to the fountain-head in
Egypt itself.
1 There ', continues the story, ' he studied with the priests
and prophets and instructed himself on every possible topic,
neglecting no item of the instruction favoured by the best
judges, no individual man among those who were famous for
their knowledge, no rite practised in the country wherever it
was, and leaving no place unexplored where he thought he
could discover something more. . . . And so he spent 22
years in the shrines throughout Egypt, pursuing astronomy
and geometry and, of set purpose and not by fits and starts or
casually, entering into all the rites of divine worship, until he
was taken captive by Cambyses's force and carried off to
Babylon, where again he consorted with the Magi, a willing
pupil of willing masters. By them he was fully instructed in
their solemn rites and religious worship, and in their midst he
attained to the highest eminence in arithmetic, music, and the
other branches of learning. After twelve years more thus
spent he returned to Samos, being then about 56 years old.'
Whether these stories are true in their details or not is
a matter of no consequence. They represent the traditional
and universal view of the Greeks themselves regarding the
beginnings of their philosophy, and they reflect throughout
the Greek spirit and outlook.
From a scientific point of view a very important advantage
possessed by the Greeks was their remarkable capacity for
accurate observation. This is attested throughout all periods,
by the similes in Homer, by vase-paintings, by the ethno-
graphic data in Herodotus, by the ' Hippocratean ' medical
books, by the biological treatises of Aristotle, and by the
history of Greek astronomy in all its stages. To take two
commonplace examples. Any person who examines the
under-side of a horse's hoof, which we call a ' frog ' and the
6 INTRODUCTORY
Greeks called a 'swallow', will agree that the latter is
the more accurate description. Or again, what exactness
of perception must have been possessed by the architects and
workmen to whom we owe the pillars which, seen from below,
appear perfectly straight, but, when measured, are found to
bulge out (eVrao-i?).
A still more essential fact is that the Greeks were a race of
thinkers. It was not enough for them to know the fact (the
otl)\ they wanted to know the why and wherefore (the 8ia tl),
and they never rested until they were able to give a rational
explanation, or what appeared to them to be such, of every
fact or phenomenon. The history of Greek astronomy fur-
nishes a good example of this, as well as of the fact that no
visible phenomenon escaped their observation. We read in
Cleomedes 1 that there were stories of extraordinary lunar
eclipses having been observed which ' the more ancient of the
mathematicians ' had vainly tried to explain ; the supposed
' paradoxical ' case was that in which, while the sun appears
to be still above the western horizon, the eclipsed moon is
seen to rise in the east. The phenomenon was seemingly
inconsistent with the recognized explanation of lunar eclipses
as caused by the entrance of the moon into the earth's
shadow ; how could this be if both bodies were above the
horizon at the same time ? The ' more ancient ' mathemati-
cians tried to argue that it was possible that a spectator
standing on an eminence of the spherical earth might see
along the generators of a cone, i.e. a little downwards on all
sides instead of merely in the plane of the horizon, and so
might see both the sun and the moon although the latter was
in the earth's shadow. Cleomedes denies this, and prefers to
regard the whole story of such cases as a fiction designed
merely for the purpose of plaguing astronomers and philoso-
phers ; but it is evident that the cases had actually been
observed, and that astronomers did not cease to work at the
problem until they had found the real explanation, namely
that the phenomenon is due to atmospheric refraction, which
makes the sun visible to us though it is actually beneath the
horizon. Cleomedes himself gives this explanation, observing
that such cases of atmospheric refraction were especially
1 Cleomedes, De motu circulari, ii. 6, pp. 218 sq.
DEVELOPMENT OF PHILOSOPHY 7
noticeable in the neighbourhood of the Black Sea, and com-
paring the well-known experiment of the ring at the bottom
of a jug, where the ring, just out of sight when the jug is
empty, is brought into view when water is poured in. We do
not know who the ' more ancient ' mathematicians were who
were first exercised by the ' paradoxical ' case ; but it seems
not impossible that it was the observation of this phenomenon,
and the difficulty of explaining it otherwise, which made
Anaxagoras and others adhere to the theory that there are
other bodies besides the earth which sometimes, by their
interposition, cause lunar eclipses. The story is also a good
illustration of the fact that, with the Greeks, pure theory
went hand in hand with observation. Observation gave data
upon which it was possible to found a theory ; but the theory
had to be modified from time to time to suit observed new
facts ; they had continually in mind the necessity of * saving
the phenomena' (to use the stereotyped phrase of Greek
astronomy). Experiment played the same part in Greek
medicine and biology.
Among the different Greek stocks the Ionians who settled
on the coast of Asia Minor were the most favourably situated
in respect both of natural gifts and of environment for initiat-
ing philosophy and theoretical science. When the colonizing
spirit first arises in a nation and fresh fields for activity and
development are sought, it is naturally the younger, more
enterprising and more courageous spirits who volunteer to
leave their homes and try their fortune in new countries ;
similarly, on the intellectual side, the colonists will be at
least the equals of those who stay at home, and, being the
least wedded to traditional and antiquated ideas, they will be
the most capable of striking out new lines. So it was with
the Greeks who founded settlements in Asia Minor. The
geographical position of these settlements, connected with the
mother country by intervening islands, forming stepping-
stones as it were from the one to the other, kept them in
continual touch with the mother country ; and at the same
time their geographical horizon was enormously extended by
the development of commerce over the whole of the Mediter-
ranean. The most adventurous seafarers among the Greeks
of Asia Minor, the Phocaeans, plied their trade successfully
8 INTRODUCTORY
as far as the Pillars of Hercules, after they had explored the
Adriatic sea, the west coast of Italy, and the coasts of the
Ligurians and Iberians. They are said to have founded
Massalia, the most important Greek colony in the western
countries, as early as 600 B.C. Cyrene, on the Libyan coast,
was founded in the last third of the seventh century. The
Milesians had, soon after 800 b. a, made settlements on the
east coast of the Black Sea (Sinope was founded in 785) ; the
first Greek settlements in Sicily were made from Euboea and
Corinth soon after the middle of the eighth century (Syracuse
734). The ancient acquaintance of the Greeks with the south
coast of Asia Minor and with Cyprus, and the establishment
of close relations with Egypt, in which the Milesians had a
large share, belongs to the time of the reign of Psammetichus I
(664-610 B.C.), and many Greeks had settled in that country.
The free communications thus existing with the whole of
the known world enabled complete information to be collected
with regard to the different conditions, customs and beliefs
prevailing in the various countries and races ; and, in parti-
cular, the Ionian Greeks had the inestimable advantage of
being in contact, directly and indirectly, with two ancient
civilizations, the Babylonian and the Egyptian.
Dealing, at the beginning of the Metaphysics, with the
evolution of science, Aristotle observes that science was
preceded by the arts. The arts were invented as the result
of general notions gathered from experience (which again was
derived from the exercise of memory) ; those arts naturally
came first which are directed to supplying the necessities of
life, and next came those which look to its amenities. It .was
only when all such arts had been established that the sciences,
which do not aim at supplying the necessities or amenities
of life, were in turn discovered, and this happened first in
the places where men began to have leisure. This is why
the mathematical arts were founded in Egypt ; for there the
priestly caste was allowed to be at leisure. Aristotle does not
here mention Babylon ; but, such as it was, Babylonian
science also was the monopoly of the priesthood.
It is in fact true, as Gomperz saysj that the first steps on
the road of scientific inquiry were, so far as we know from
1 Griechische Denker, i, pp. 36, 37.
DEVELOPMENT OF PHILOSOPHY 9
history, never accomplished except where the existence of an
organized caste of priests and scholars secured the necessary
industry, with the equally indispensable continuity of tradi-
tion. But in those very places the first steps were generally
the last also, because the scientific doctrines so attained tend,
through their identification with religious prescriptions, to
become only too easily, like the latter, mere lifeless dogmas.
It was a fortunate chance for the unhindered spiritual de-
velopment of the Greek people that, while their predecessors
in civilization had an organized priesthood, the Greeks never
had. To begin with, they could exercise with perfect freedom
their power of unerring eclecticism in the assimilation of every
kind of lore. ' It remains their everlasting glory that they
discovered and made use of the serious scientific elements in
the confused and complex mass of exact observations and
superstitious ideas which constitutes the priestly wisdom of
the East, and threw all the fantastic rubbish on one side.' 1
For the same reason, while using the earlier work of
Egyptians and Babylonians as a basis, the Greek genius
could take an independent upward course free from every
kind of restraint and venture on a flight which was destined
to carry it to the highest achievements.
The Greeks then, with their ' unclouded clearness of mind '
and their freedom of thought, untrammelled by any ' Bible ' or
its equivalent, were alone capable of creating the sciences as
they did create them, i.e. as living things based on sound first
principles and capable of indefinite development. It was a
great boast, but a true one, which the author of the Epinomis
made when he said, ' Let us take it as an axiom that, whatever
the Greeks take from the barbarians, they bring it to fuller
perfection '. 2 He has been speaking of the extent to which
the Greeks had been able to explain the relative motions and
speeds of the sun, moon and planets, while admitting that
there was still much progress to be made before absolute
certainty could be achieved. He adds a characteristic sen-
tence, which is very relevant to the above remarks about the
Greek's free outlook :
' Let no Greek ever be afraid that we ought not at any time
to studv things divine because we are mortal. We ought to
1 Cumont, Neue Jahrbiicher, xxiv, 1911, p. 4. 2 Epinomis, 987 D.
10 INTRODUCTORY
maintain the very contrary view, namely, that God cannot
possibly be without intelligence or be ignorant of human
nature : rather he knows that, when he teaches them, men
will follow him and learn what they are taught. And he is
of course perfectly aware that he does teach us, and that we
learn, the very subject we are now discussing, number and
counting; if he failed to know this, he would show the
greatest want of intelligence ; the God we speak of would in
fact not know himself, if he took it amiss that a man capable
of learning should learn, and if he did not rejoice unreservedly
with one who became good by divine influence.' 1
Nothing could well show more clearly the Greek conviction
that there could be no opposition between religion and scien-
tific truth, and therefore that there could be no impiety in the
pursuit of truth. The passage is a good parallel to the state-
ment attributed to Plato that deb? dd yeco/zeryoer.
Meaning and classification of mathematics.
The words pady/iaTa and /za^Ty/zari/coy do not appear to
have been definitely appropriated to the special meaning of
mathematics and mathematicians or things mathematical until
Aristotle's time. With Plato jiddrjixa is quite general, mean-
ing any subject of instruction or study; he speaks of kccXo,
HaOrjjxara^ good subjects of instruction, as of KaXd €7tlttj8€v-
fiara, good pursuits, of women's subjects as opposed to men's,
of the Sophists hawking sound ^laOrjixara ; what, he asks in
the Republic, are the greatest fxaOrjfjLaTa 1 and he answers that
the greatest fiddrjfia is the Idea of the Good. 2 But in the
Laivs he speaks of Tpia yLaOrwiara^ three subjects, as fit for
freeborn men, the subjects being arithmetic, the science of
measurement (geometry), and astronomy 3 ; and no doubt the"
pre-eminent place given to mathematical subjects in his scheme
of education would have its effect in encouraging the habit of
speaking of these subjects exclusively as fiaOrjiiara. The
Peripatetics, we are told, explained the special use of the
word in this way ; they pointed out that, whereas such things
as rhetoric and poetry and the whole of popular fjiovcriKrj can
be understood even by one who has not learnt them, the sub-
jects called by the special name of fiaOrj/xaTa cannot be known
1 Epinomis, 988 a. 2 Republic, vi. 505 A. 3 Laws, vii. 817 E.
CLASSIFICATION OF MATHEMATICS 11
by any one who has not first gone through a course of instruc-
tion in them ; they concluded that it was for this reason that
these studies were called \iaQr]ixa7iKr\} The special use of the
word /MaOrj/jLaTLKrj seems actually to have originated in the
school of Pythagoras. It is said that the esoteric members
of the school, those who had learnt the theory of know-
ledge in its most complete form and with all its elaboration
of detail, were known as iiaBrmaTiKoi, mathematicians (as
opposed to the dKovorfiariKoi, the exoteric learners who were
entrusted, not with the inner theory, but only with the prac-
tical rules of conduct) ; and, seeing that the Pythagorean
philosophy was mostly mathematics, the term might easily
come to be identified with the mathematical subjects as
distinct from others. According to Anatolius, the followers
of Pythagoras are said to have applied the term fjLadrj/xaTiKrj
more particularly to the two subjects of geometry and
arithmetic, which had previously been known by their own
separate names only and not by any common designation
covering both. 2 There is also an apparently genuine frag-
ment of Archytas, a Pythagorean and a contemporary and
friend of Plato, in which the word fxadrinara appears as
definitely appropriated to mathematical subjects :
1 The mathematicians (tol irepl tcc jxadrjfj.ara) seem to me to
have arrived at correct conclusions, and it is not therefore
surprising that they have a true conception of the nature of
each individual thing : for, having reached such correct con-
clusions regarding the nature of the universe, they were
bound to see in its true light the nature of particular things
as well. Thus they have handed down to us clear knowledge
about the speed of the stars, their risings and settings, and
about geometry, arithmetic, and sphaeric, and last, not least,
about music ; for these fxaBrjiiara seem to be sisters/ 3
This brings us to the Greek classification of the different
branches of mathematics. Archytas, in the passage quoted,
specifies the four subjects of the Pythagorean quadrivium,
geometry, arithmetic, astronomy, and music (for ' sphaeric '
means astronomy, being the geometry of the sphere con-
1 Anatolius in Hultsch's Heron, pp. 276-7 (Heron, vol. iv, Heiberg,
p. 160. 18-24).
2 Heron, ed. Hultsch, p. 277 ; vol. iv, p. 160. 24-162. 2, Heiberg.
3 Diels, Vorsokratiker, i 3 , pp. 330-1.
12 INTRODUCTORY
sidered solely with reference to the problem of accounting for
the motions of the heavenly bodies) ; the same list of subjects
is attributed to the Pythagoreans by Nicomachus, Theon of
Smyrna, and Proclus, only in a different order, arithmetic,
music, geometry, and sphaeric ; the idea in this order was
that arithmetic and music were both concerned with number
(ttcktov), arithmetic with number in itself, music with number
in relation to something else, while geometry and sphaeric were
both concerned with magnitude (ir-qXiKov), geometry with mag-
nitude at rest, sphaeric with magnitude in motion. In Plato's
curriculum for the education of statesmen the same subjects,
with the addition of stereometry or solid geometry, appear,
arithmetic first, then geometry, followed by solid geometry,
astronomy, and lastly harmonics. The mention of stereometry
as an independent subject is Plato's own idea ; it was, however,
merely a formal addition to the curriculum, for of course
solid problems had been investigated earlier, as a part of
geometry, by the Pythagoreans, Democritus and others.
Plato's reason for the interpolation was partly logical. Astro-
nomy treats of the motion of solid bodies. There is therefore
a gap between plane geometry and astronomy, for, after con-
sidering plane figures, we ought next to add the third dimen-
sion and consider solid figures in themselves, before passing
to the science which deals with such figures in motion. But
Plato emphasized stereometry for another reason, namely that
in his opinion it had not been sufficiently studied. ' The
properties of solids do not yet seem to have been discovered.'
He adds :
' The reasons for this are two. First, it is because no State
holds them in honour that these problems, which are difficult,
are feebly investigated ; and, secondly, those who do investi-
gate them are in need of a superintendent, without whose
guidance they are not likely to make discoveries. But, to
begin with, it is difficult to find such a superintendent, and
then, even supposing him found, as matters now stand, those
who are inclined to these researches would be prevented by
their self-conceit from paying any heed to him.' 1
I have translated o>9 vvv 'iyzi (' as matters now stand ') in
this passage as meaning 'in present circumstances', i.e. so
1 Plato, Republic, vii. 528 A-c.
CLASSIFICATION OF MATHEMATICS 13
long as the director has not the authority of the State behind
him : this seems to be the best interpretation in view of the
whole context ; but it is possible, as a matter of construction,
to connect the phrase with the preceding words, in which case
the meaning would be ' and, even when such a superintendent
has been found, as is the case at present', and Plato would
be pointing to some distinguished geometer among his con-
temporaries as being actually available for the post. If Plato
intended this, it would presumably be either Archytas or
Eudoxus whom he had in mind.
It is again on a logical ground that Plato made harmonics
or music follow astronomy in his classification. As astronomy
is the motion of bodies (cjyopd fidQovs) and appeals to the eye,
so there is a harmonious motion (evapiiovios (popd), a motion
according to the laws of harmony, which appeals to the ear.
In maintaining the sisterhood of music and astronomy Plato
followed the Pythagorean view (cf. the passage of Archytas
above quoted and the doctrine of the ' harmony of the
spheres ').
(a) Arithmetic and logistic.
By arithmetic Plato meant, not arithmetic in our sense, but
the science which considers numbers in themselves, in other
words, what we mean by the Theory of Numbers. He does
not, however, ignore the art of calculation (arithmetic in our
sense); he speaks of number and calculation {dpiQfibv kol
Xoyicr/xov) and observes that ' the art of calculation (XoytorTLKrj)
and arithmetic (dpiOp.r)TLKrj) are both concerned with number';
those who have a natural gift for calculation (ol (pxxrei Xoyi-
(ttikoi) have, generally speaking, a talent for learning of all
kinds, and even those who are slow are, by practice in it,
made smarter. 1 But the art of calculation (Xoyio-TiKrj) is only
preparatory to the true science ; those who are to govern the
city are to get a grasp of XoyKrriKrj, not in the popular
sense with a view to use in trade, but only for the purpose of
knowledge, until they are able to contemplate the nature of
number in itself by thought alone. 2 This distinction between
dpiOfirjTiKrj (the theory of numbers) and XoyiariKT] (the art of
1 Republic, vii. 522 c, 525 a, 526 b.
2 lb. vii. 525 b, c.
14 INTRODUCTORY
calculation) was a fundamental one in Greek mathematics.
It is found elsewhere in Plato, 1 and it is clear that it was well
established in Plato's time. Archytas too has Xoyto-TiKr] in
the same sense ; the art of calculation, he says, seems to be far
ahead of other arts in relation to wisdom or philosophy, nay
it seems to make the things of which it chooses to treat even
clearer than geometry does ; moreover, it often succeeds even
where geometry fails. 2 But it is later writers on the classification
of mathematics who alone go into any detail of what Xoyiari ktj
included. Geminus in Proclus, Anatolius in the Variae Collec-
tiones included in Hultsch's Heron, and the scholiast to Plato's
Charmides are our authorities. Arithmetic, says Geminus, 3 is
divided into the theory of linear numbers, the theory of plane
numbers, and the theory of solid numbers. It investigates,
in and by themselves, the species of number as they are succes-
sively evolved from the unit, the formation of plane numbers,
similar and dissimilar, and the further progression to the third
dimension. As for the XoyLo-TiKo?, it is not in and by themselves
that he considers the properties of numbers but with refer-
ence to sensible objects; and for this reason he applies to
them names adapted from the objects measured, calling some
(numbers) firjXiTtjs (from firjXov, a sheep, or firjXou, an apple,
more probably the latter) and others (piaXirrjs (from (pidXr),
a bowl). 4 The scholiast to the Charmides is fuller still : 5
' Logistic is the science which deals with numbered things,
not numbers ; it does not take number in its essence,
but it presupposes 1 as unit, and the numbered object as
number, e.g. it regards 3 as a triad, 10 as a decad, and
applies the theorems of arithmetic to such (particular) cases.
Thus it is logistic which investigates on the one hand what
Archimedes called the cattle-problem, and on the other hand
melites and pldalites numbers, the latter relating to bowls,
the former to flocks (he should probably have said " apples ") ;
in other kinds too it investigates the numbers of sensible
bodies, treating them as absolute (coy nepl reXeieov). Its sub-
ject-matter is everything that is numbered. Its branches
include the so-called Greek and Egyptian methods in multi-
plications and divisions, 6 the additions and decompositions
1 Cf. Gorgias, 451 B, c ; Theaetetus, 145 A with 198 A, &c.
2 Diels, Vorsokratiker, i s , p. 337. 7-11.
3 Proclus on Eucl. I, p. 39. 14-20. 4 lb., p. 40. 2-5.
5 On Charmides, 165 e. 6 See Chapter II, pp. 52-60.
ARITHMETIC AND LOGISTIC 15
of fractions ; which methods it uses to explore the secrets of
the theory of triangular and polygonal numbers with reference
to the subject-matter of particular problems.'
The content of logistic is for the most part made fairly
clear by the scholia just quoted. First, it comprised the
ordinary arithmetical operations, addition, subtraction, multi-
plication, division, and the handling of fractions ; that is, it
included the elementary parts of what we now call arithmetic.
Next, it dealt with problems about such things as sheep
(or apples), bowls, &c. ; and here we have no difficulty in
recognizing such problems as we find in the arithmetical
epigrams included in the Greek anthology. Several of them
are problems of dividing a number of apples or nuts among
a certain number of persons ; others deal with the weights of
bowls, or of statues and their pedestals, and the like ; as a
rule, they involve the solution of simple equations with one
unknown, or easy simultaneous equations with two unknowns;
two are indeterminate equations of the first degree to be solved
in positive integers. From Plato's allusions to such problems
it is clear that their origin dates back, at least, to the fifth
century B.C. The cattle-problem attributed to Archimedes
is of course a much more difficult problem, involving the
solution of a ( Pellian ' equation in numbers of altogether
impracticable size. In this problem the sums of two pairs
of unknowns have to be respectively a square and a tri-
angular number; the problem would therefore seem to *
correspond to the description of those involving ' the theory
of triangular and polygonal numbers '. Tannery takes the
allusion in the last words to be to problems in indeter-
minate analysis like those of Diophantus's Arithmetica. The
difficulty is that most of Diophantus's problems refer to num-
bers such that their sums, differences, &c, are squares, whereas
the scholiast mentions only triangular and polygonal numbers.
Tannery takes squares to be included among polygons, or to
have been accidentally omitted by a copyist. But there is
only one use in Diophantus's Arithmetica of a triangular
number (in IV. 38), and none of a polygonal number ; nor can
the Tptyoovovs of the scholiast refer, as Tannery supposes, to
right-angled triangles with sides in rational numbers (the
main subject of Diophantus's Book VI), the use of the mascu-
16 INTRODUCTORY
line showing that only rpiyoovovs dpiOfiovs, triangular num-
bers, can be meant. Nevertheless there can, I think, be no
doubt that Diophantus's Arithmetica belongs to Logistic.
Why then did Diophantus call his thirteen books Arithmetica ?
The explanation is probably this. Problems of the Diophan-
tine type, like those of the arithmetical epigrams, had pre-
viously been enunciated of concrete numbers (numbers of
apples, bowls, &c), and one of Diophantus's problems (V. 30)
is actually in epigram form, and is about measures of wine
with prices in drachmas. Diophantus then probably saw that
there was no reason why such problems should refer to
numbers of any one particular thing rather than another, but
that they might more conveniently take the form of finding
numbers in the abstract with certain properties, alone or in
combination, and therefore that they might claim to be part
of arithmetic, the abstract science or theory of numbers.
It should be added that to the distinction between arith-
metic and logistic there corresponded (up to the time of
Nicomachus) different methods of treatment. With rare
exceptions, such as Eratosthenes's kovkivov, or sieve, a device
for separating out the successive prime numbers, the theory
of numbers was only treated in connexion with geometry, and
for that reason only the geometrical form of proof was used,
whether the figures took the form of dots marking out squares,
triangles, gnomons, &c. (as with the early Pythagoreans), or of
straight lines (as in Euclid VII-IX) ; even Nicomachus did
not entirely banish geometrical considerations from his work,
and in Diophantus's treatise on Polygonal Numbers, of which
a fragment survives, the geometrical form of proof is used.
(/?) Geometry and geodaesia.
By the time of Aristotle there was separated out from
geometry a distinct subject, yeoc>8aL<ria, geodesy, or, as we
should say, mensuration, not confined to land -measuring, but
covering generally the practical measurement of surfaces and
volumes, as we learn from Aristotle himself, 1 as well as from
a passage of Geminus quoted by Proclus. 2
1 Arist. Metaph. B. 2, 997 b 26, 31.
2 Proclus on Eucl. I, p. 39. 20-40. 2.
PHYSICAL SUBJECTS AND THEIR BRANCHES 17
(y) Physical subjects, mechanics, optics, harmonics,
astronomy, and their branches.
In applied mathematics Aristotle recognizes optics and
mechanics in addition to astronomy and harmonics. He calls
optics, harmonics, and astronomy the more physical (branches)
of mathematics, 1 and observes that these subjects and mechanics
depend for the proofs of their propositions upon the pure
mathematical subjects, optics on geometry, mechanics on
geometry or stereometry, and harmonics on arithmetic ; simi-
larly, he says, Phaenomena (that is, observational astronomy)
depend on (theoretical) astronomy. 2
The most elaborate classification of mathematics is that given
by Geminus. 3 After arithmetic and geometry, which treat of
non-sensibles, or objects of pure thought, come the branches
which are concerned with sensible objects, and these are six
in number, namely mechanics, astronomy, optics, geodesy,
canonic (kclvov acq), logistic. Anatolius distinguishes the same
subjects but gives them in the order logistic, geodesy, optics,
canonic, mechanics, astronomy. 4 Logistic has already been
discussed. Geodesy too has been described as mensuration,
the practical measurement of surfaces and volumes; as
Geminus says, it is the function of geodesy to measure, not
a cylinder or a cone (as such), but heaps as cones, and tanks
or pits as cylinders. 5 Canonic is the theory of the musical
intervals as expounded in works like Euclid's KaraTo/irj
kclvovos, Division of the canon*
Optics is divided by Geminus into three branches. 6 (1) The
first is Optics proper, the business of which is to explain why
things appear to be of different sizes or different shapes
according to the way in which they are placed and the
distances at which they are seen. Euclid's Optics consists
mainly of propositions of this kind; a circle seen edge-
wise looks like a straight line (Prop. 22), a cylinder seen by
one eye appears less than half a cylinder (Prop. 28); if the
line joining the eye to the centre of a circle is perpendicular
1 Arist. Fhys. ii. 2, 194 a 8.
2 Arist. Anal. Post. i. 9, 76 a 22-5 ; i. 13, 78 b 35-9.
3 Proclus on Eucl. I, p. 38. 8-12.
4 See Heron, ed. Hultsch, p. 278 ; ed. Heiberg, iv, p. 164.
5 Proclus on Eucl. I, p. 39. 23-5. 6 lb., p. 40. 13-22.
1523 C
18 INTRODUCTORY
to the plane of the circle, all its diameters will look equal
(Prop. 34), but if the joining line is neither perpendicular to
the plane of the circle nor equal to its radius, diameters with
which it makes unequal angles will appear unequal (Prop. 35) ;
if a visible object remains stationary, there exists a locus such
that, if the eye is placed at any point on it, the object appears
to be of the same size for every position of the eye (Prop. 38).
(2) The second branch is Catoptric, or the theory of mirrors,
exemplified by the Catoptrica of Heron, which contains,
e. g., the theorem that the angles of incidence and reflexion
are equal, based on the assumption that the broken line
connecting the eye and the object reflected is a minimum.
(3) The third branch is o-KrjvoypacpiKrj or, as we might say,
scene-painting, i.e. applied perspective.
Under the general term of mechanics Geminus 1 dis-
tinguishes (1) opy avoir oukt), the art of making engines of war
(cf. Archimedes's reputed feats at the siege of Syracuse and
Heron's /3eAo7roiiVca), (2) Oav/j.aToiTouKrj, the art of making
wonderful machines, such as those described in Heron's
Pneumatica and Automatic Theatre, (3) Mechanics proper,
the theory of centres of gravity, equilibrium, the mechanical
powers, &c, (4) Sphere-making, the imitation of the move-
ments of the heavenly bodies ; Archimedes is said to have
made such a sphere or orrery. Last of all, 2 astronomy
is divided into (1) ypoofj,oi>iKrj, the art of the gnomon, or the
measurement of time by means of the various forms of
sun-dials, such as those enumerated by Vitruvius, 3 (2) /xerecopo-
arKOTTiKrj, which seems to have included, among other things,
the measurement of the heights at which different stars cross
the meridian, (3) SioTTTpiKr}, the use of the dioptra for the
purpose of determining the relative positions of the sun,
moon, and stars.
Mathematics in Greek education. 4
The elementary or primary stage in Greek education lasted
till the age of fourteen. The main subjects were letters
(reading and writing followed by dictation and the study of
1 Proclus on Eucl. I, p. 41. 3-18. 2 lb., pp. 41. 19-42. 6.
3 Vitruvius, De architecture/,, ix. 8.
4 Cf. Freeman, Schools of Hellas, especially pp. 100-7, 159.
MATHEMATICS IN GREEK EDUCATION 19
literature), music and gymnastics ; but there is no reasonable
doubt that practical arithmetic (in our sense), including
weights and measures, was taught along with these subjects.
Thus, at the stage of spelling, a common question asked of
the pupils was, How many letters are there in such and such
a word, e.g. Socrates, and in what order do they come ? 1 This
would teach the cardinal and ordinal numbers. In the same
connexion Xenophon adds, ' Or take the case of numbers.
Some one asks, What is twice five?' 2 This indicates that
counting was a part of learning letters, and that the multipli-
cation table was a closely connected subject. Then, again,
there were certain games, played with cubic dice or knuckle-
bones, to which boys were addicted and which involved some
degree of arithmetical skill. In the game of knucklebones in
the Lysis of Plato each boy has a large basket of them, and
the loser in each game pays so many over to the winner. 3
Plato connects the art of playing this game with mathe-
matics 4 ; so too he associates irtTTeia (games with ireo-croi,
somewhat resembling draughts or chess) with arithmetic in
general. 5 When in the Laws Plato speaks of three subjects
fit for freeborn citizens to learn, (1) calculation and the science
of numbers, (2) mensuration in one, two and three dimen-
sions, and (3) astronomy in the sense of the knowledge of
the revolutions of the heavenly bodies and their respective
periods, he admits that profound and accurate knowledge of
these subjects is not for people in general but only for a few. 6
But it is evident that practical arithmetic was, after letters
and the lyre, to be a subject for all, so much of arithmetic,
that is, as is necessary for purposes of war, household
management, and the work of government. Similarly, enough
astronomy should be learnt to enable the pupil to understand
the calendar. 7 Amusement should be combined with instruc-
tion so as to make the subjects attractive to boys. Plato was
much attracted by the Egyptian practice in this matter : 8
* Freeborn boys should learn so much of these things as
vast multitudes of boys in Egypt learn along with their
1 Xenophon, Econ. viii. 14. 2 Xenophon, Mem. iv. 4. 7.
3 Plato, Lysis, 206 e ; cf. Apollonius Rhodius, hi. 117.
4 Phaedrus, 274 c-d. 5 Politicus, 299 e ; Laws, 820 c.
6 Laws, 817 e-818 A. 7 lb. 809 c, d.
8 lb. 819 a-c.
c 2
20 INTRODUCTORY
letters. First there should be calculations specially devised
as suitable for boys, which they should learn with amusement
and pleasure, for example, distributions of apples or garlands
where the same number is divided among more or fewer boys,
or (distributions) of the competitors in boxing or wrestling
matches on the plan of drawing pairs with byes, or by taking
them in consecutive order, or in any of the usual ways 1 ; and
again there should be games with bowls containing gold,
bronze, and silver (coins'?) and the like mixed together, 2 or the
bowls may be distributed as undivided units ; for, as I said,
by connecting with games the essential operations of practical
arithmetic, you supply the boy with what will be useful to
him later in the ordering of armies, marches and campaigns,
as well as in household management ; and in any case you
make him more useful to himself and more wide awake.
Then again, by calculating measurements of things which
have length, breadth, and depth, questions on all of which
the natural condition of all men is one of ridiculous and dis-
graceful ignorance, they are enabled to emerge from this
state.'
It is true that these are Plato's ideas of what elementary
education should include ; but it can hardly be doubted that
such methods were actually in use in Attica.
Geometry and astronomy belonged to secondary education,
which occupied the years between the ages of fourteen and
eighteen. The pseudo-Platonic Axiochus attributes to Prodi-
cus a statement that, when a boy gets older, i. e. after he has
1 The Greek of this clause is (t)iavop.a\) ttvktcdv ko\ naXaiaTcov icpedpcias
tc Kai (rvW^ecos iv pipei Ka\ icpe^rjs Kai as ireCpvKacn yiyveadai. So far as
I can ascertain, iv pipei (by itself) and i(pe£rjs have always been taken
as indicating alternative methods, ' in turn and in consecutive order '.
But it is impossible to get any satisfactory contrast of meaning between
' in turn ' and ' in consecutive order \ It is clear to me that we have
here merely an instance of Plato's habit of changing the order of words
for effect, and that iv pipci must be taken with the genitives i(pet)pet,as Kai
avWrjgeojs ; i.e. w.e must translate as if we had iv icpedpeias re ko! crvWrj-
£eoos p,ipei, ' by way o/byes and drawings '. This gives a proper distinction
between (1) drawings with byes and (2) taking competitors in consecutive
order.
2 It is difficult to decide between the two possible interpretations
of the phrase (piaXas apa xpuo-ov Kai x a ^ K °^ Kai opyvpov Kai toiovtow tivcov
aXXcov Kfpawvvres. It may mean ' taking bowls made of gold, bronze,
silver and other metals mixed together (in certain proportions) ' or
* filling bowls with gold, bronze, silver, &c. (sc. objects such as coins)
mixed together '. The latter version seems to agree best with rraifoires
(making a game out of the process) and to give the better contrast to
4 distributing the bowls as wholes' (okas na>s diadiduvres).
MATHEMATICS IN GREEK EDUCATION 21
passed the primary stage under the paidagogos, grammatistes,
and paidotribes, he comes under the tyranny of the [ critics ',
the geometers, the tacticians, and a host of other masters. 1
Teles, the philosopher, similarly, mentions arithmetic and
geometry among the plagues of the lad. 2 It would appear
that geometry and astronomy were newly introduced into the
curriculum in the time of Isocrates. ' I am so far ', he says, 3
' from despising the instruction which our ancestors got, that
I am a supporter of that which has been established in our
time, I mean geometry, astronomy, and the so-called eristic
dialogues.' Such studies, even if they do no other good,
keep the young out of mischief, and in Isocrates's opinion no
other subjects could have been invented more useful and
more fitting ; but they should be abandoned by the time that
the pupils have reached man's estate. Most people, he says,
think them idle, since (say they) they are of no use in private
or public affairs ; moreover they are forgotten directly because
they do not go with us in our daily life and action, nay, they
are altogether outside everyday needs. He himself, however,
is far from sharing these views. True, those who specialize in
such subjects as astronomy and geometry get no good from
them unless they choose to teach them for a livelihood ; and if
they get too deeply absorbed, they become unpractical and
incapable of doing ordinary business ; but the study of these
subjects up to the proper point trains a boy to keep his atten-
tion fixed and not to allow his mind to wander ; so, being
practised in this way and having his wits sharpened, he will be
capable of learning more important matters with greater ease
and speed. Isocrates will not give the name of ' philosophy ' to
studies like geometry and astronomy, which are of no imme-
diate use for producing an orator or man of business ; they
are rather means of training the mind and a preparation for
philosophy. They are a more manly discipline than the sub-
jects taught to boys, such as literary study and music, but in
other respects have the same function in making them quicker
to learn greater and more important subjects.
1 Axiochus, 366 E.
2 Stobaeus, Eel iv. 34, 72 (vol. v, p. 848, 19 sq., Wachsmuth and
Hense).
3 See Isocrates, Panathenaicus, S§ 26-8 (238 b-d) : Uep\ dvTi86cr(ai?,
§§ 261-8.
22 INTRODUCTORY
It would appear therefore that, notwithstanding the in-
fluence of Plato, the attitude of cultivated people in general
towards mathematics was not different in Plato's time from
what it is to-day.
We are told that it was one of the early Pythagoreans,
unnamed, who first taught geometry for money : ' One of the
Pythagoreans lost his property, and when this misfortune
befell him he was allowed to make money by teaching
geometry.' 1 We may fairly conclude that Hippocrates of
Chios, the first writer of Elements, who also made himself
famous by his quadrature of lunes, his reduction of the
duplication of the cube to the problem of finding two mean
proportionals, and his proof that the areas of circles are in
the ratio of the squares on their diameters, also taught for
money and for a like reason. One version of the story is that
he was a merchant, but lost all his property through being
captured by a pirate vessel. He then came to Athens to
prosecute the offenders and, during a long stay, attended
lectures, finally attaining such proficiency in geometry that
he tried to square the circle. 2 Aristotle has the different
version that he allowed himself to be defrauded of a large
sum by custom-house officers at Byzantium, thereby proving,
in Aristotle's opinion, that, though a good geometer, he was
stupid and incompetent in the business of ordinary life. 3
We find in the Platonic dialogues one or two glimpses of
mathematics being taught or discussed in school- or class-
rooms. In the Erastae 4 Socrates is represented as going into
the school of Dionysius (Plato's own schoolmaster 5 ) and find-
ing two lads earnestly arguing some point of astronomy ;
whether it was Anaxagoras or Oenopides whose theories they
were discussing he could not catch, but they were drawing
circles and imitating some inclination or other with their
hands. In Plato's Theaetetus 6 we have the story of Theodoras
lecturing on surds and proving separately, for the square root
of every non-square number from 3 to 17, that it is incom-
mensurable with 1, a procedure which set Theaetetus and the
1 Iamblichus, Vit. Pyth. 89.
2 Philoponus on Arist. Phys., p. 327 l> 44-8, Brandis.
3 Eudemian Ethics, H. 14, 1247 a 17.
4 Erastae, 32 a, b. 5 Diog. L. iii. 5.
6 Theaetetus, 147 D-148 B.
MATHEMATICS IN GREEK EDUCATION 23
younger S^prates thinking whether it was not possible to
comprehend all such surds under one definition. In these two
cases we have advanced or selected pupils discussing among
themselves the subject of lectures they had heard and, in the
second case, 'trying to develop a theory of a more general
character.
But mathematics was not only taught by regular masters
in schools ; the Sophists, who travelled from place to place
giving lectures, included mathematics (arithmetic, geometry,
and astronomy) in their very wide list of subjects. Theo-
doras, who was Plato's teacher in mathematics and is
described by Plato as a master of geometry, astronomy,
logistic and music (among other subjects), was a pupil of
Protagoras, the Sophist, of Abdera. 1 Protagoras himself, if we
may trust Plato, did not approve of mathematics as part of
secondary education ; for he is made to say that
1 the other Sophists maltreat the young, for, at an age when
the young have escaped the arts, they take them against their
will and plunge them once more into the arts, teaching them
the art of calculation, astronomy, geometry, and music — and
here he cast a glance at Hippias — whereas, if any one comes
to me, he will not be obliged to learn anything except what
he comes for.' 2
The Hippias referred to is of course Hippias of Elis, a really
distinguished mathematician, the inventor of a curve known
as the quaclratrix which, originally intended for the solution
of the problem of trisecting any angle, also served (as the
name implies) for squaring the circle. In the Hippias Minor 3
there is a description of Hippias's varied accomplishments.
He claimed, according to this passage, to have gone once to
the Olympian festival with everything that he wore made by
himself, ring and seal (engraved), oil- bottle, scraper, shoes,
clothes, and a Persian girdle of expensive type ; he also took
poems, epics, tragedies, dithyrambs, and all sorts of prose
works. He was a master of the science of calculation
(logistic), geometry, astronomy, ' rhythms and harmonies
and correct writing'. He also had a wonderful system of
mnemonics enabling him, if he once heard a string of fifty
1 Theaetetus, 164 e, 168 e. 2 Protagoras, 318 d, e.
3 Hippias Minor, pp. 366 c-368 e.
24 INTRODUCTORY
names, to remember them all. As a detail, we are told that
he got no fees for his lectures in Sparta, and that the Spartans
could not endure lectures on astronomy or geometry or
logistic; it was only a small minority of them who could
even count ; what they liked was history and archaeology.
The above is almost all that we know of the part played
by mathematics in the Greek system of education. Plato's
attitude towards mathematics was, as we have seen, quite
exceptional ; and it was no doubt largely owing to his influence
and his inspiration that mathematics and astronomy were so
enormously advanced in his school, and especially by Eudoxus
of Cnidos and Heraclides of Pontus. But the popular atti-
tude towards Plato's style of lecturing was not encouraging.
There is a story of a lecture of his on ' The Good ' which
Aristotle was fond of telling. 1 The lecture was attended by
a great crowd, and ' every one went there with the idea that
he would be put in the way of getting one or other of* the
things in human life which are usually accounted good, such
as Riches, Health, Strength, or, generally, any extraordinary
gift of fortune. But' when they found that Plato discoursed
about mathematics, arithmetic, geometry, and astronomy, and
finally declared the One to be the Good, no wonder they were
altogether taken by surprise ; insomuch that in the end some
of the audience were inclined to scoff at the whole thing, while
others objected to it altogether.' Plato, however, was able to
pick and choose his pupils, and he could therefore insist on
compliance with the notice which he is said to have put over
his porch, ' Let no one unversed in geometry enter my doors ' ; 2
and similarly Xenocrates, who, after Speusippus, succeeded to
the headship of the school, could turn away an applicant for
admission who knew no geometry with the words ' Go thy
way, for thou hast not the means of getting a grip of
philosophy \ 3
The usual attitude towards mathematics is illustrated by
two stories of Pythagoras and Euclid respectively. Pytha-
goras, we are told, 4 anxious as he was to transplant to his own
country the system of education which he had seen in opera-
1 Aristoxenus, Harmonica, ii ad hi it.
2 Tzetzes, Chiliad, viii. 972. s Diog. L. iv. 10.
4 Iambi ichus, Vit. Pyth. c. 5.
MATHEMATICS IN GREEK EDUCATION 25
tion in Egypt, and the study of mathematics in particular,
could get none of the Samians to listen to him. He adopted
therefore this plan of communicating his arithmetic and
geometry, so that it might not perish with him. Selecting
a young man who from his behaviour in gymnastic exercises
seemed adaptable and was withal poor, he promised him that,
if he would learn arithmetic and geometry systematically, he
would give him sixpence for each ' figure ' (proposition) that he
mastered. This went on until the youth got interested in
the subject, when Pythagoras rightly judged that he would
gladly go on without the sixpence. He therefore hinted
that he himself was poor and must try to earn his daily bread
instead of doing mathematics ; whereupon the youth, rather
than give up the study, volunteered to pay sixpence himself
to Pythagoras for each proposition. We must presumably
connect with this story the Pythagorean motto, ' a figure and
a platform (from which to ascend to the next higher step), not
a figure and sixpence '.*
The other story is that of a pupil who began to learn
geometry with Euclid and asked, when he had learnt one
proposition, ' What advantage shall I get by learning these
things ? ' And Euclid called the slave and said, ' Give him
sixpence, since he must needs gain by what he learns.'
We gather that the education of kings in the Macedonian
period did not include much geometry, whether it was Alex-
ander who asked Menaechmus, or Ptolemy who asked Euclid,
for a short-cut to geometry, and got the reply that ' for travel-
ling over the country there are royal roads and roads for com-
mon citizens : but in geometry there is one road for all \ 2
1 Proclus on Eucl. I, p. 84. 16.
2 Stobaeus, Eel. ii. 31, 115 (vol. ii, p. 228, 30, Wachsmuth).
II
GREEK NUMERICAL NOTATION AND ARITH-
METICAL OPERATIONS
The decimal system.
The Greeks, from the earliest historical times, followed the
decimal system of numeration, which had already been
adopted by civilized peoples all the world over. There are,
it is true, traces of quinary reckoning (reckoning in terms of
five) in very early times ; thus in Homer Trefnrdgeiv (to ' five ')
is used for ' to count \ l But the counting by fives was pro-
bably little more than auxiliary to counting by tens ; five was
a natural halting-place between the unit and ten, and the use
of five times a particular power of ten as a separate category
intermediate between that power and the next was found
convenient in the earliest form of numerical symbolism estab-
lished in Greece, just as it was in the Roman arithmetical
notation. The reckoning by Rve does not amount to such a
variation of the decimal system as that which was in use
among the Celts and Danes; these peoples had a vigesimal
system, traces of which are still left in the French quatre-
vingts, quatre-vingt-treize, &c, and in our score, three-score
and ten, twenty-one, &c.
The natural explanation of the origin of the decimal system,
as well as of the quinary and vigesimal variations, is to
suppose that they were suggested by the primitive practice of
reckoning with the fingers, first of one hand, then of both
together, and after that with the ten toes in addition (making
up the 20 of the vigesimal system). The subject was mooted
in the Aristotelian Problems, 2 where it is asked :
{ Why do all men, whether barbarians or Greeks, count up
to ten, and not up to any other number, such as 2, 3, 4, or 5,
so that, for example, they do not say one-^us-five (for 6),
1 Homer, Od. iv. 412. 2 xv. 3, 910 b 23-911 a 4.
THE DECIMAL SYSTEM 27
two-p/us-five (for 7), as they say one-plus-ten (eV#e/ca, for 11),
two-plus-ten (ScoSeKa, for 12), while on the other hand they
do not go beyond ten for the first halting-place from which to
start again repeating the units? For of course any number
is the next before it plus 1, or the next before that plus 2,
and so with those preceding numbers ; yet men fixed definitely
on ten as the number to count up to. It cannot have been
chance ; for chance will not account for the same thing being
done always : what is always and universally done is not due
to chance but to some natural cause.'
Then, after some fanciful suggestions (e.g. that 10 is a
' perfect number '), the author proceeds :
' Or is it because men were born with ten fingers and so,
because they possess the equivalent of pebbles to the number
of their own fingers, come to use this number for counting
everything else as well ? '
Evidence for the truth of this latter view is forthcoming in
the number of cases where the word for 5 is either the same
as, or connected with, the word for ' hand \ Both the Greek
\€ip and the Latin manus are used to denote ' a number ' (of
men). The author of the so-called geometry of Boetius says,
moreover, that the ancients called all the numbers below ten
by the name digits (' fingers 'J. 1
Before entering on a description of the Greek numeral signs
it is proper to refer briefly to the systems of notation used
by their forerunners in civilization, the Egyptians and
Babylonians.
Egyptian numerical notation.
The Egyptians had a purely decimal system, with the signs
I for the unit, n for 10, @ for 100, f for 1,000, ] for 10,000,
^^ for 100,000. The number of each denomination was
expressed by repeating the sign that number of times ; when
the number was more than 4 or 5, lateral space was saved by
arranging them in two or three rows, one above the other.
The greater denomination came before the smaller. Numbers
could be written from left to right or from right to left ; in
the latter case the above signs were turned the opposite way.
The fractions in use were all submultiples or single aliquot
1 Boetius, Be Inst. Ar., &c, p. 395. 6-9, Fnedlein.
28 GREEK NUMERICAL NOTATION
parts, except § , which had a special sign <:L> or < rp > ; the
submultiples were denoted by writing «^=> over the corre-
sponding whole number ; thus
mi xx
2S nnn
Babylonian systems.
(a) Decimal, (ft) Sexagesimal.
The ancient Babylonians had two systems of numeration.
The one was purely decimal based on the following signs.
The simple wedge T represented the unit, which was repeated
up to nine times : where there were more than three, they
were placed in two or three rows, e.g. ^ = 4, yij =7. 10
was represented by ^; 11 would therefore be /Y . 100 had
the compound sign T>-, and 1000 was expressed as 10 hun-
dreds, by ^J*% the prefixed ^ (10) being here multiplicative.
Similarly, the ^T*~ was regarded as one sign, and /^T*- de-
noted not 2000 but 10000, the prefixed \ being again multi-
plicative. Multiples of 10000 seem to have been expressed
as multiples of 1000: at least, 120000 seems to be attested
in the form 100.1000 + 20.1000. The absence of any definite
unit above 1000 (if it was really absent) must have rendered
the system very inconvenient as a means of expressing large
numbers.
Much more interesting is the second Babylonian system,
the sexagesimal. This is found in use on the Tables of
Senkereh, discovered by W. K. Loftus in 1854, which may go
back as far as the time between 2300 and 1600 B.C. In this
system numbers above the units (which go from 1 to 59) are
arranged according to powers of 60. 60 itself was called
sussu ( = soss), 60 2 was called sar, and there was a name also
(ner) for the intermediate number 10.60 = 600. The multi-
ples of the several powers of 60, 60 2 , 60 3 , &c, contained in the
number to be written down were expressed by means of the
same wedge-notation as served for the units, and the multi-
pies were placed in columns side by side, the columns being
appropriated to the successive powers of 60. The unit-term
EGYPTIAN AND BABYLONIAN NOTATION 29
was followed by similar columns appropriated, in order, to the
successive submultiples — , — h s &c, the number of sixtieths,
1 60 60 2
&c, being again denoted by the ordinary wedge-numbers.
Thus %^ <(<^ $< represents 44.60 2 + 26.60 + 40 = 160,000;
44ffl W 46(ffi = 27 ' 6 ° 2+ 2L60 + 36 = 98,496. Simi-
larly we find ^^ ^^ representing 30 + §£ and ^^ KKKT^YT
representing 30+§J; the latter case also shows that the
Babjdonians, on occasion, used the subtractive plan, for the 2 7
is here written 30 minus 3.
The sexagesimal system only required a definite symbol
for (indicating the absence of a particular denomination),
and a fixed arrangement of columns, to become a complete
position-value system like the Indian. With a sexagesimal
system would occur comparatively seldom, and the Tables of
Senkereh do not show a case ; but from other sources it
appears that a gap often indicated a zero, or there was a sign
used for the purpose, namely i, called the 'divider'. The
inconvenience of the system was that it required a multipli-
cation table extending from 1 times 1 to 59 times 59. It had,
however, the advantage that it furnished an easy means of
expressing very large numbers. The researches of H. V.
Hilprecht show that 60 4 = 12,960,000 played a prominent
part in Babylonian arithmetic, and he found a table con-
taining certain quotients of the number T^
= 60 8 +10.60 7 , or 195,955,200,000,000. Since the number of
units of any denomination are expressed in the purely decimal
notation, it follows that the latter system preceded the sexa-
gesimal. What circumstances led to the adoption of 60 as
the base can only be conjectured, but it may be presumed that
the authors of the system were fully alive to the convenience
of a base with so many divisors, combining as it does the
advantages of 12 and 10.
Greek numerical notation.
To return to the Greeks. We find, in Greek inscriptions of
all dates, instances of numbers and values written out in full ;
but the inconvenience of this longhand, especially in such
things as accounts, would soon be felt, and efforts would be
made to devise a scheme for representing numbers more
30 GREEK NUMERICAL NOTATION
concisely by means of conventional signs of some sort. The
Greeks conceived the original idea of using the letters of the
ordinary Greek alphabet for this purpose.
(a) The ' Herodianic ' signs.
There were two main systems of numerical notation in use in
classical times. The first, known as the Attic system and
used for cardinal numbers exclusively, consists of the set of
signs somewhat absurdly called ' Herodianic ' because they are
described in a fragment 1 attributed to Herodian, a gram-
marian of the latter half of the second century A.D. The
authenticity of the fragment is questioned, but the writer
says that he has seen the signs used in Solon's laws, where
the prescribed pecuniary fines were stated in this notation,
and that they are also to be found in various ancient inscrip-
tions, decrees and laws. These signs cannot claim to be
numerals in the proper sense ; they are mere compendia or
abbreviations; for, except in the case of the stroke I repre-
senting a unit, the signs are the first letters of the full words
for the numbers, and all numbers up to 50000 were repre-
sented by combinations of these signs. I, representing the
unit, may be repeated up to four times ; P (the first letter of
Trivre) stands for 5, A (the first letter of SeKa) for 10, H
(representing Zkcltov) for 100, X (ylXioi) for 1000, and M
(fivpioi) for 10000. The half-way numbers 50, 500, 5000
were expressed by combining P (five) with the other signs
respectively; F, F, P, made up of P (5) and A (10), = 50;
F, made up of P and H, = 500 ; F = 5000 ; and P™ = 5Q000.
There are thus six simple and four compound symbols, and all
other numbers intermediate between those so represented are
made up by juxtaposition on an additive basis, so that each
of the simple signs may be repeated not more than four times ;
the higher numbers come before the lower. For example,
PI =6, All II = 14, HP = 105, XXXXFHHHHFAAAAPMII
= 4999. Instances of this system of notation are found in
Attic inscriptions from 454 to about 95 B.C. Outside Attica
the same system was in use, the precise form of the symbols
varying with the form of the letters in the local alphabets.
Thus in Boeotian inscriptions f 1 or P~ = 50, f-E= 100, OE =500,
1 Printed in the Appendix to Stephanus's Thesaurus, vol. viii.
THE 'HERODIANIC SIGNS 31
V = 1000, 4P=5000; and ^HE HE hE hEW>! 1 1 = 5823. But,
in consequence of the political influence of Athens, the Attic
system, sometimes with unimportant modifications, spread to
other states. 1
In a similar manner compendia were used to denote units
of coinage or of weight. Thus in Attica T — rdXavrov (6000
drachmae), M = jj.i>a (1000 drachmae), Z or $ = o-rarrip
(1/3 00 0th of a talent or 2 drachmae), h = Spaxp-ij, I = 6(3o\6s
(l/6th of a drachma), C = rjfjiioofiiXioi' (l/l2th of a drachma),
3 or T = TerapTrj/j.opioi' (l/4th of an obol or 1/2 4th of a
drachma), X = x a ^Kovs (l/8th of an obol or 1/4 8th of a
drachma). Where a number of one of these units has to be
expressed, the sign for the unit is written on the left of that
for the number; thus hPAl '= 61 drachmae. The two com-
pendia for the numeral and the unit are often combined into
one ; e.g. fP, F 1 = 5 talents, p = 50 talents, H = 100 talents,
P^= 500 talents, £= 1000 talents, A = 10 minas, P = 5 drach-
mae, A, A, ^= 10 staters, &c.
(f3) The ordinary alphabetic numerals.
The second main system, used for all kinds of numerals, is
that with which we are familiar, namely the alphabetic
system. The Greeks took their alphabet from the Phoe-
nicians. The Phoenician alphabet contained 22 letters, and,
in appropriating the different signs, the Greeks had the
happy inspiration to use for the vowels, which were not
written in Phoenician, the signs for certain spirants for which
the Greeks had no use ; Aleph became A, He was used for E,
Yod for I, and Ayin for O ; when, later, the long E was
differentiated, Cheth was used, B or H. Similarly they
utilized superfluous signs for sibilants. Out of Zayin and
Samech they made the letters Z and H. The remaining two
sibilants were Ssade and Shin. From the latter came the
simple Greek 2 (although the name Sigma seems to corre-
spond to the Semitic Samech, if it is not simply the ' hissing '
letter, from crifa). Ssade, a softer sibilant ( = <r<r), also called
San in early times, was taken over by the Greeks in the
place it occupied after n, and written in the form M or v\,
The form T ( = crcr) appearing in inscriptions of Halicarnassus
1 Larfeld, Handbuch der griechischen Epigrajpihik, vol. i, p. 417.
32
GREEK NUMERICAL NOTATION
(e.g. 'AXiKapvaTleoov] = *A\iKOLpvallk<£>v) andTeos ([O^aXdTrjs ;
cf. BdXaiiav in another place) seems to be derived from some
form of Ssade ; this T, after its disappearance from the
literary alphabet, remained as a numeral, passing through
the forms /l\, m, F 1 , fl>, and <p to the fifteenth century form \
to which in the second half of the seventeenth century the
name Sampi was applied (whether as being the San which
followed Pi or from its resemblance to the cursive form of tt).
The original Greek alphabet also retained the Phoenician Vau (F)
in its proper place between E and Z and the Koppa = Qoph (9)
immediately before P. The Phoenician alphabet ended with
T ; the Greeks first added T, derived from Vau apparently
(notwithstanding the retention of F), then the letters 4>, X, ^
and, still later, 12. The 27 letters used for numerals are
divided into three sets of nine each ; the first nine denote
the units, 1, 2, 3, &c, up to 9 ; the second nine the tens, from
10 to 90; and the third nine the hundreds, from 100 to 900.
The following is the scheme :
A = 1 I = 10
B = 2 K = 20
r =3 A = 30
A = 4 M = 40
E = 5 N = 50
C [9] = 6 S = 60
Z = 7 O = 70
h =8 n = 80
6=9 9 = 90
The sixth sign in the first column (L) is a form of the
digamma F F. It came, in the seventh and eighth centuries
A. D., to be written in the form Cj and then, from its similarity
to the cursive 9 (= <tt), was called Stigma.
This use of the letters of the alphabet as numerals was
original with the Greeks ; they did not derive it from the
Phoenicians, who never used their alphabet for numerical
purposes but had separate signs for numbers. The earliest
occurrence of numerals written in this way appears to be in
a Halicarnassian inscription of date not long after 450 B.C.
Two caskets from the ruins of a famous mausoleum built at
Halicarnassus in 351 B.C., which are attributed to the time
of Mausolus, about 350 B.C., are inscribed with the letters
p
= 100
2
= 200
T
= 300
Y
= 400
= 500
X
= 600
*
= 700
n
= 800
TP>]
= 900
THE ORDINARY ALPHABETIC NUMERALS 33
*NA = 754 and Z<?r = 293. A list of priests of Poseidon
at Halicarnassus, attributable to a date at least as early as the
fourth century, is preserved in a copy of the second or first
century, and this copy, in which the numbers were no doubt
reproduced from the original list, has the terms of office of the
several priests stated on the alphabetical system. Again, a
stone inscription found at Athens and perhaps belonging to
the middle of the fourth century B.C. has, in five fragments
of columns, numbers in tens and units expressed on the same
system, the tens on the right and the units on the left.
There is a difference of opinion as to the approximate date
of the actual formulation of the alphabetical system of
numerals. According to one view, that of Larfeld, it must
have been introduced much earlier than the date (450 B.C. or
a little later) of the Halicarnassus inscription, in fact as early
as the end of the eighth century, the place of its origin being
Miletus. The argument is briefly this. At the time of the
invention of the system all the letters from A to H, including
F and 9 in their proper places, were still in use, while
Ssade (T, the double ss) had dropped out; this is why the
last-named sign (afterwards ~^) was put at the end. If
C (= 6) and 9 (= 90 ) na( l been no longer in use as letters,
they too would have been put, like Ssade, at the end. The
place of origin of the numeral system must have been one in
which the current alphabet corresponded to the content and
order of the alphabetic numerals. The order of the sign's
<t>, X, "¥ shows that it was one of the Eastern group of
alphabets. These conditions are satisfied by one alphabet,
and one only, that of Miletus, at a stage which still recognized
the Vau (F) as well as the Koppa (9). The 9 is found along
with the so-called complementary letters including CI, the
latest of all, in the oldest inscriptions of the Milesian colony
Naucratis (about 650 b. c); and, although there are no
extant Milesian inscriptions containing the F, there is at all
events one very early example of F in Ionic, namely 'Aya-
o-iXeFo (AyacnXijFov) on a vase in the Boston (U.S.) Museum
of Fine Arts belonging to the end of the eighth or (at latest)
the middle of the seventh century. Now, as D. is fully
established at the date of the earliest inscriptions at Miletus
(about 700 B.C.) and Naucratis (about 650 B.C.), the earlier
1523 D
34 GREEK NUMERICAL NOTATION
extension of the alphabet by the letters <t> X y must have
taken place not later than 750 B.C. Lastly, the presence in
the alphabet of the Vau indicates a time which can hardly
be put later than 700 B.C. The conclusion is that it was
about this time, if not earlier, that the numerical alphabet
was invented.
The other view is that of Keil, who holds that it originated
in Dorian Caria, perhaps at Halicarnassus itself, about
550-425 B.C., and that it was artificially put together by
some one who had the necessary knowledge to enable him
to fill up his own alphabet, then consisting of twenty-four
letters only, by taking over F and 9 from other alphabets and
putting them in their proper places, while he completed the
numeral series by adding T at the end. 1 Keil urges, as
against Larfeld, that it is improbable that F and n ever
existed together in the Milesian alphabet. Larfeld' s answer 2
is that, although F had disappeared from ordinary language
at Miletus towards the end of the eighth century, we cannot
say exactly when it disappeared, and even if it was practically
gone at the time of the formulation of the numerical alphabet,
it would be in the interest of instruction in schools,, where
Homer was read, to keep the letter as long as possible in the'
official alphabet. On the other hand, Keil's argument is open
to the objection that, if the Carian inventor could put the
F and 9 into their proper places in the series, he would hardly
have failed to put the Ssade T in its proper place also, instead
of at the end, seeing that T is found in Caria itself, namely
in a Halicarnassus (Lygdamis) inscription of about 453 B.C.,
and also in Ionic Teos about 476 k.c 3 (see pp. 31-2 above).
It was a long time before the alphabetic numerals found
general acceptance. They were not officially used until the
time of the Ptolemies, when it had become the practice to write,
in inscriptions and on coins, the year of the reign of the ruler
for the time being. The conciseness of the signs made them
particularly suitable for use on coins, where space was limited.
When coins went about the world, it was desirable that the
notation should be uniform, instead of depending on local
alphabets, and it only needed the support of some paramount
1 Hermes, 29, 1894, p. 265 sq. 2 Larfeld, op. cit.,.i, p. 421.
3 lb., i, p. 358.
THE ORDINARY ALPHABETIC NUMERALS 35
political authority to secure theffinal triumph of the alphabetic
system. The alphabetic numerals are found at Alexandria
on coins of Ptolemy II, Philadelphus, assigned to 266 B.C.
A coin with the inscription 'A\e£dv8pov KA (twenty-fourth
year after Alexander's death) belongs, according to Keil, to
the end of the third century. 1 A very old Graeco-Egyptian
papyrus (now at Leyden, No. 397), ascribed to 257 B.C. ;
contains the number kO = 29. While in Boeotia the Attic
system was in use in the middle of the third century, along
with the corresponding local system, it had to give way about
200 B.C. to the alphabetic system, as is shown by an inventory
from »the temple of Amphiaraus at Oropus 2 ; we have here
the first official use of the alphabetic system in Greece proper.
From this time Athens stood alone in retaining the archaic
system, and had sooner or later to come into line with other
states. The last certainly attested use of the Attic notation
in Athens was about 95 B. c. ; the alphabetic numerals were
introduced there some time before 50 B.C., the first example
belonging to the time of Augustus, and by a.d. 50 they were
in official use.
The two systems are found side by side in a number of
papyrus-rolls found at Herculaneum (including the treatise
of Philodemus De pietate, so that the rolls cannot be older than
40 or 50 B.C.); these state on the title page, after the name of
the author, the number of books in alphabetic numerals, and
the number of lines in the Attic notation, e.g. ETTIKpYPOY
TTEPI | 0YIEHI | IE dpiO . . XXXHH (where IE = 15 and
XXXHH = 3200), just as we commonly use Roman figures,
to denote Books and Arabic figures for sections or lines. 3
1 Hermes, 29, 1894, p. 276 n.
2 Keil in Hermes, 25, 1890, pp. 614-15.
3 Reference should be made, in passing, to another, g«as/-numerical,
use of the letters of the ordinary alphabet, as current at the time, for
numbering particular things. As early as the fifth century we find in
a Locrian bronze-inscription the letters A to © (including f then and
there current) used to distinguish the nine paragraphs of the text. At
the same period the Athenians, instead of following the old plan of
writing out ordinal numbers in full, adopted the more convenient device
of denoting them by the letters of the alphabet. In the oldest known
example opos K indicated 'boundary stone No. 10 ' ; and in the fourth
century the tickets of the ten panels of jurymen were marked with the
letters A to K. In like manner the Books in certain works of Aristotle
(the Ethics, Metaphysics, Politics, and 'Topics) were at some time
D 2
36 GREEK NUMERICAL NOTATION
(y) Mode of writing numbers in the ordinary alphabetic
notation.
Where, in the alphabetical notation, the number to be
written contained more than one denomination,, say, units
with tens, or with tens and hundreds, the higher numbers
were, as a rule, put before the lower. This was generally the
case in European Greece ; on the other hand, in the inscrip-
tions of Asia Minor, the smaller number comes first, i. e. the
letters are arranged in alphabetical order. Thus 111 may be
represented either by P I A or by A I P ; the arrangement is
sometimes mixed, as PA I. The custom of writing the numbers
in descending order became more firmly established in later
times through the influence of the corresponding Roman
practice. 1
The alphabetic numerals sufficed in themselves to express
all numbers from 1 to 999. For thousands (up to 9000) the
letters were used again with a distinguishing mark ; this was
generally a sloping stroke to the left, e.g. 'A or ,A = 1000,
but other forms are also found, e.g. the stroke might be
combined with the letter as A = 1000 or again f A= 1000,
C C = 6000. For tens of thousands the letter M (fj.vpioi) was
borrowed from the other system, e.g. 2 myriads would be
B
BM, MB, or M.
To distinguish letters representing numbers from the
letters of the surrounding text different devices are used:
sometimes the number is put between dots j or : , or separ-
ated by spaces from the text on both sides of it. In Imperial
times distinguishing marks, such as a horizontal stroke above
the letter, become common, e.g. rj fiovXr) tS>v X, other
variations being •X, X% X and the like.
In the cursive writing with which we are familiar the
numbered on the same principle ; so too the Alexandrine scholars
(about 280 B.C.) numbered the twenty-four Books of Homer with the
letters A to 12. When the number of objects exceeded 24, doubled
letters served for continuing- the series, as AA, BB, &c. For example,
a large quantity of building-stones have been found -, among these are
stones from the theatre at the Piraeus marked AA, BB, &c, and again
AA|BB, BB|BB, &c. when necessary. Sometimes the numbering by
double letters was on a different plan, the letter A denoting the full
number of the first set of letters (24) ; thus AP would be 24+17 =41.
1 Larfeld. op. cit., i, p. 426.
ORDINARY ALPHABETIC NOTATION 37
orthodox way of distinguishing numerals was by a horizontal
stroke above each sign or collection of signs ; the following
was therefore the scheme (with 5- substituted for F repre-
senting 6, and with "^ — 900 at the end) :
units (1 to 9) a, 0, y, 8, e,j, £, jj,9;
tens (10 to 90) I, k, A, /x, P, £, 0, -fir, 9j_
hundreds (100 to 900) p, cr, f, v, 0, x> i/r, a>, "^ ;
thousands (1000 to 9000) ,a, ,£, /y, ,§, ,e, ,<-, ,£ ,ij, ,0;
(for convenience of printing, the horizontal stroke above the
sign will hereafter, as a rule, be omitted).
(8) Comparison of the two systems of numerical notation.
The relative merits of the two systems of numerical
notation used by the Greeks have been differently judged.
It will be observed that the m^a/-numerals correspond
closely to the Roman" numerals, except that there is no
formation of numbers by subtraction as IX, XL, XC; thus
XXXXHHHHHFAAAAPIIII = MMMMDCCCCLXXXX VI 1 1 1
as compared with MMMMCMXCI X = 4999. The absolute
inconvenience of the Roman system will be readily appreci-
ated by any one who has tried to read Boetius (Boetius
would write the last-mentioned number aslV.DCCCCXCVIIII).
Yet Cantor l draws a comparison between the two systems
much to the disadvantage of the alphabetic numerals.
{ Instead ', he says, ' of an advance we have here to do with
a decidedly retrograde step, especially so far as its suitability
for the further development of the numeral system is con-
cerned. If we compare the older "Herodianic" numerals
with the later signs which we have called alphabetic numerals,
we observe in the latter two drawbacks which do not attach
to the former. There now had to be more signs, with values
to be learnt by heart ; and to reckon with them required
a much greater effort of memory. The addition
AAA + AAAA = PAA (30 + 40 = 70)
could be coordinated in one act of memory with that of
HHH + HHHH = FHH (300 + 400 = 700)
in so far as the sum of 3 and 4 units of the same kind added
1 Cantor, Gesch. d. Math. I 3 , p. 129.
38 GREEK NUMERICAL NOTATION
up to 5 and 2 units of the same kind. On the other hand
A + [i = o did not at all immediately indicate that r + v = yjr.
The new notation had only one advantage over the other,
namely that it took less space. Consider, for instance, 849,
which in the " Herodianic " form is FHHHAAAAPIIII, but
in the alphabetic system is couO. The former is more self-
explanatory and, for reckoning with, has most important
advantages.' Gow follows Cantor, but goes further and says
that ' the alphabetical numerals were a fatal mistake and
hopelessly confined such nascent arithmetical faculty as the
Greeks may have possessed ' ! 1 On the other hand, Tannery,
holding that the merits of the alphabetic numerals could only
be tested by using them, practised himself in their use until,
applying them to the whole of the calculations in Archimedes's
Measurement of a Circle, he found that the alphabetic nota-
tion had practical advantages which he had hardly suspected
before, and that the operations took little longer with Greek
than with modern numerals. 2 Opposite as these two views are,
they seem to be alike based on a misconception. Surely we do
not ' reckon with ' the numeral signs at all, but with the
^uords for the numbers which they -represent. For instance,
in Cantor's illustration, we do not conclude that the figure 3
and the figure 4 added together make the figure 7 ; what we
do is to say ' three and four are seven '. Similarly the Greek
would not say to himself ' y and 8 = £ ' but rpeis Kal ria-a-ape?
{tttol ; and, notwithstanding what Cantor says, this would
indicate the corresponding addition ' three hundred and four
hundred are seven hundred ', rpiaKOcrioi Kal TerpaKoa-ioL
eTrraKoaioi, and similarly with multiples of ten or of 1000 or
10000. Again, in using the multiplication table, we say
1 three times four is twelve ', or ' three multiplied by four =
twelve ' ; the Greek would say Tph revo-apes, or rpeis kirl
reo-crapas, SdoSeKa, and this would equally indicate that ' thirty
times forty is twelve hundred or one thousand two hundred ',
or that ' thirty times four hundred is tivelve thousand or a
myriad and two thousand' (rpiaKovTOLKLS TeacrapaKoura x^ioi
Kal SiaKoaiot, or rpiaKovT&Kis TerpaKoo-ioi fivpioi Kal 8i<r)(i\ioi).
1 Gow, A Short History of Greek Mathematics, p. 46.
2 Tannery, Menwires scimtifiques (ed. Heiberg and Zeuthen), i,
pp. 200-1.
COMPARISON OF THE TWO SYSTEMS 39
The truth is that in mental calculation (whether the opera-
tion be addition, subtraction, multiplication, or division), we
reckon with the corresponding words, not with the symbols,
and it does not matter a jot to the calculation how we choose
to write the figures down. While therefore the alphabetical
numerals had the advantage over the ' Herodianic ' of being
so concise, their only disadvantage was that there were more
signs (twenty -seven) the meaning of which had to be com-
mitted to memory : truly a very slight disadvantage. The
one real drawback to the alphabetic system was the absence
of a sign for (zero) ; for the for ovSe/xia or ovSev which
we find in Ptolemy was only used in the notation of sexa-
gesimal fractions, and not as part of the numeral system. If
there had been a sign or signs to indicate the absence in
a number of a particular denomination, e. g. units or tens or
hundreds, the Greek symbols could have been made to serve
as a position-value system scarcely less effective than ours.
For, while the position-values are clear in such a number
as 7921 (,£~Va), it would only be necessary in the case of
such a number as 7021 to show a blank in the proper place
by writing, say, ,£- Ka. Then, following Diophantus's plan
of separating any number of myriads by a dot from the
thousands, &c, we could write ^^/ca . ^tttS for 79216384 or
X . - t - 5 for 70000304, while we could continually add
sets of four figures to the left, separating each set from the
next following by means of a dot.
(e) Notation for large numbers.
Here too the orthodox way of writing tens of thousands
was by means of the letter M with the number of myriads
above it, e.g. M = 20000, M ^cooe — 71755875 (Aristarchus
Y
of Samos) ; another method was to write M or M for the
myriad and to put the number of myriads after it, separated
by a dot from the remaining thousands, &c, e. g.
Y
M/)i/./V£= 1507984
(Diophantus, IV. 28). Yet another way of expressing myriads
was to use the symbol representing the number of myriads
with two dots over it; thus a^cpqP = 18592 (Heron, Geo-
metrica, 17. 33). The word fivpidSe? could, of course, be
40 GREEK NUMERICAL NOTATION
written in full, e.g. fivptdBes fivor) kcli ^)i(3 — 22780912
(ib. 17. 34). To express still higher numbers, powers of
myriads were used; a myriad (10000) was a first myriad
(rrpodTT) jivpids) to distinguish it from a second myriad (Sevripa
fivpid?) or 10000 2 , and so on; the words npcoTcu pvpidSes,
SevT€pai pvpidStsj &c., could either be written in full or
Y Y
expressed by M, MM, &c, respectively; thus Sevrepai fivpidSts
o
f<7 npooTai (pvpidSes) fi~^vq M /^ = 16 2958 6560 (Dio
o
phantus, V. 8), where M = povaSes (units) is inserted to
distinguish the fi^ivq, the number of the ' first myriads ',
from the /j"0£ denoting 6560 units.
(i) Apollonius's ' tetrads '.
The latter system is the same as that adopted by Apollonius
in an arithmetical work, now lost, the character of which is,
however, gathered from the elucidations in Pappus, Book II ;
the only difference is that Apollonius called his tetrads (sets
of four digits) pvpiaSes anXaT, SnrXat, rpnrXaT, &c, 'simple
myriads', 'double', 'triple', &c, meaning 10000, 10000 2 ,
10000 3 , and so on. The abbreviations for these successive
powers in Pappus are p. a , p , pF, &c. ; thus p? jev^fi kcu fi fi /y\
teal p° fiv = 5462 3600 6400 0000. Another, but a less con-
venient, method of denoting the successive powers of 10000
is indicated by Nicolas Rhabdas (fourteenth century A.D.)
who says that, while a pair of dots above the ordinary
numerals denoted the number of myriads, the ' double
myriad ' was indicated by two pairs of dots one above the other,
the ' triple myriad ' by three pairs of dots, and so on. Thus
"^ = 9000000, $ = 2 (10000) 2 , pi = 40 (10000) 3 , and so on.
(ii) Archimedes's system (by octads).
Yet another special system invented for the purpose of
expressing very large numbers is that of Archimedes's
Psammites or Sand-reckoner. This goes by octads :
10000 2 = 100000000 = 10 8 ,
and all the numbers from 1 to 10 8 form the first order;
the last number, 10 8 , of the first order is taken as the unit
of the second order, which consists of all the numbers from
ARCHIMEDES'S SYSTEM (BY OCTADS) 41
10 s , or 100000000, to 10 16 , or 100000000 2 ; similarly 10 16 is
taken as the unit of the third order, which consists of all
numbers from 10 lu to 10 24 , and so on, the 100000000th order
consisting of all the numbers from (lOOOOOOOO) 99999999 to
(100000000) 100000000 , i.e. from lO 8 ^ 10 " 1 ) to 10 8 - 10 '. The aggre-
gate of all the orders up to the 100000000th form the first
period; that is, if P = (lOOOOOOOO) 10 ", the numbers of the
first period go from 1 to P. Next, P is the unit of the first
order of the second period ; the first order of the second
period then consists of all numbers from P up to 100000000 P
or P.10 8 ; P.10 8 is the unit of the second order (of the
second period) which ends. with (100000000) 2 P or P.10 16 ;
P. 10 1G begins the third order of the second 'period, and so
on ; the 100000000th order of the second period consists of
the numbers from (100000000) 999999j9 P or P . 10 8 ^ 10 - 1 ) to
(100000000) 100000000 P or P. lO 810 *, i.e. P 2 . Again, P 2 is the
unit of the first order of the third period, and so on. The
first order of the 100000000th period consists of the numbers
from P 10 _1 to P 10 -1 . 10 8 , the second order of the same
period of the numbers from P 10 *- 1 . 10 8 to P 10_1 . 10 16 , and so
on, the (10 8 )th order of the (10 8 )th period, or the period
itself, ending with P 10 *- 1 . lo 810 *, i.e. P 10 \ The last number
is described by Archimedes as a ' myriad- myriad units of the
myriad-myriaclth order of the myriad-myriadth period (at
fjLVpiaKKrflVpLOO-Tds 7T€pl68oV fJ.VpiaKL<7/J.VpLO(TTCOJ/ CCpiOflCOV /IVplCU
livpi&Ses) '. This system was, however, a tour de force, and has
nothing to do with the ordinary Greek numerical notation.
Fractions.
(a) The Egyptian system.
We now come to the methods of expressing fractions. A
fraction may be either a submultiple (an ' aliquot part ', i. e.
a fraction with numerator unity) or an ordinary proper
fraction with a number not unity for numerator and a
greater number for denominator. The Greeks had a pre-
ference for expressing ordinary proper fractions as the sum
of two or more submultiples ; in this they followed the
Egyptians, who always expressed fractions in this way, with
the exception that they had a single sign for §, whereas we
42 GREEK NUMERICAL NOTATION
should have expected them to split it up into J + i, as f was
split up into J + i- The orthodox sign for a submultiple
was the letter for the corresponding number (the denomi-
nator) but with an accent instead of a horizontal stroke
above it; thus y r = -J, the full expression being y /lipos =
rpLTov fiepos, a third part (y' is in fact short for Tpiros, so
that it is also used for the ordinal number ' third ' as well
as for the fraction J, and similarly with all other accented
numeral signs) ; A/3' — J^, pi/3' — T ^, &c. There were
special signs for J, namely 1/ or C', 1 and for •§, namely w r .
When a number of submultiples are written one after the
other, the sum of them is meant, and similarly when they
follow a whole number ; e.g. U' 8' = \ \ or | (Archimedes) ;
k6 or' iy' \6' = 29§ A A - 29§ + A+A 01 * 29 T§;
(Heron, Geom. 15. 8, 13). But iy' to iy r means ^th times
A or lie (ibid. 12. 5), &c. A less orthodox method found
in later manuscripts was to use two accents and to write,
e.g., £" instead of {', for i. In Diophantus we find a different
mark in place of the accent ; Tannery considers the genuine
form of it to be \ so that y* = §, and so on.
(P) The ordinary Greek form, variously written.
An ordinary proper fraction (called by Euclid p,eprj, 'parts,
in the plural, as meaning a certain number of aliquot parts,
in contradistinction to fiipos, part, in the singular, which he
restricts to an aliquot part or submultiple) was expressed in
various ways. The first was to use the ordinary cardinal
number for the numerator followed by the accented number
representing the denominator. Thus we find "in Archimedes
I oa — yx an d ^ooXrj 6 la — 1838 T 9 T : (it should be noted,
however, that the I oa is a correction from oia, and this
oa
seems to indicate that the original reading was c, which
would accord with Diophantus's and Heron's method of
writing fractions). The method illustrated by these cases is
open to objection as likely to lead to confusion, since i oa!
1 It has been suggested that the forms £ and 3 for \ found in
inscriptions may perhaps represent half an O, the sign, at all events
in Boeotia, for 1 obol.
FRACTIONS 43
would naturally mean lO^V and 6 ia 9 T \ ; the context alone
shows the true meaning. Another form akin to that just
mentioned was a little less open to misconstruction ; the
numerator was written in full with the accented numeral
(for the denominator) following, e.g. Svo pt for 2/45ths
(Aristarchus of Samos). A better way was to turn the
aliquot part into an abbreviation for the ordinal number
with a termination superposed to represent the case, e.g.
<TV = | (Dioph. Lemma to V. 8), v kj™ = |§ (ibid. I. 23),
p<a v t a<£>\§V = 1834^/121 (ibid. IV. 39), just as y 09 was
written for the ordinal rpiros (cf. to s- ", the -|th part, Dioph.
IV. 39; ai'poo ra ty a 'I remove the 13ths', i.e. I multiply up
by the denominator 13, ibid. IV. 9). But the trouble was
avoided by each of two other methods.
(1) The accented letters representing the denominator were
written twice, along with the cardinal number for the
numerator. This method is mostly found in the Geometrica
and other works of Heron : cf. e Ly' iy f — T 5 g-, ra <$ £g — f .
The fractional signification is often emphasized by adding
the word X^irrd ('fractions' or 'fractional parts'), e.g. in
Ae7rra iy' iy' iff = ^§ (Geom. 12. 5), and, where the expression
contains units as well as fractions, the word ' units ' (povdSts)
is generally added, for clearness' sake, to indicate the integral
number, e.g. povdBts iff ical XewTa iy iy i/3 = 12f| (Geom.
12. 5), povdSe? ppS XeTTTct iy iy ctqO = 144- 2 T \ 9 - (Geom. 12. 6).
Sometimes in Heron fractions are alternatively given in this
notation and in that of submultiples, e.g. /? y' ie' tJtol (3 kou
& e' e' = <2j T \ or 2§ ' (Geom. 12. 48) ; { i' 1' ie' oe' JJtol
povdSesCe'e'yKal P e' e'reW e' = ' 1\^ A ts ™ 7 * + ***''
i.e. 7- + -^ (ibid.) ; rj 1/ 1 /ce' tJtol povdSes 77 e' e y kclI e to e' =
' 8 4to A or Sf + ixi', i.e. 8I + 2V (ibid. 12. 46). (In
Hultsch's edition of Heron single accents were used to de-
note whole numbers and the numerators of fractions, while
aliquot parts or denominators were represented by double
accents ; thus the last quoted expression was written
7} b 1 Ke tjtol povades r] e e y koli e to e .)
But (2) the most convenient notation of all is that which
is regularly employed by Diophantus, and occasionally in the
Metrica of Heron. In this system the numerator of any
fraction is written in the line, with the denominator above it,
44 GREEK NUMERICAL NOTATION
without accents or other mark* (except where the numerator
or denominator itself contains an accented fraction) ; the
method is therefore simply the reverse of ours, but equally
convenient. In Tannery's edition of Diophantus a line is
put between the numerator below and the denominator above :
^ 121
thus pKa = — — . But it is better to omit the horizontal line
pklJ 100
(cf. p = — — in Kenyon's Papyri ii, No. cclxv. 40, and the
1 Jo
fractions in Schone's edition of Heron's Metrica). A few
^ 2456
more instances from Diophantus may be given : pvvq —
R SWi
(IV. 28) ; frv-n = 5358 (V. 9) ; tttOW = 389 i. The deno-
v " ' 10201 v ; 152
minator is rarely found above the numerator, but to the
_ 5 15
right (like an exponent) ; e.g. te = — (I. 39). Even in the
case of a submultiple, where, as we have said, the orthodox
method was to omit the numerator and simply write the
denominator with an accent, Diophantus often follows the
(Pl/3
method applicable to other fractions, e.g. he writes a for
5T2 (IV. 28). Numbers partly integral and partly fractional,
where the fraction is a submultiple or expressed as the sum
of submultiples, are written much as we write them, the
fractions simply following the integer, e.g. a y ^ = 1 J ;
P V 9* = 2J \ (Lemma to V. 8) ; to V «r X = 370J T l (HI. 11).
Complicated fractions in which the numerator and denomi-
nator are algebraical expressions or large numbers are often
expressed by writing the numerator first and separating it
by fiopiov or kv fioptco from the denominator ; i.e. the fraction
is expressed as the numerator divided by the denominator :
Y
thus Mpv . ,f»w8 iiopiov K? . fippS = 1507984/262144 (IV. 28).
(y) Sexagesimal fractions.
Great interest attaches to the system of sexagesimal
fractions (Babylonian in its origin, as we have seen) which
was used by the Greeks in astronomical calculations, and
SEXAGESIMAL FRACTIONS 45
appears fully developed in the Syntaxis of Ptolemy,. The
circumference of a circle, and with it the four right angles
subtended by it at the centre, were divided into 360 parts
(Tfirjuara or fioipcu), as we should say degrees, each fiolpa
into 60 parts called (wpcoTa) igrjKoo-rd, (first) sixtieths or
minutes (Ae7rra),each of these again into 60 Sevrepa i£rjKOcrTd,
seconds, and so on. In like manner, the diameter of the
circle was divided into 120 Tp.rjpara, segments, and each of
these segments was divided into sixtieths, each sixtieth
again into sixty parts, and so on. Thus a convenient
fractional system was available for arithmetical calculations
in general ; for the unit could be chosen at will, and any
mixed number could be expressed as so many of those units
plus so many of the fractions which we should represent
by -go, so many of those which we should write (Vo) 2 ' (eV) 3 '
and so on to any extent. The units, rp,rjp,aTa or fiolpai (the
latter often denoted by the abbreviation fi°), were written
first, with the ordinary numeral representing the number
of them ; then came a simple numeral with one accent repre-
senting that number of first sixtieths, or minutes, then a
numeral with two accents representing that number of
second sixtieths, or seconds, and so on. Thus u° /? = 2°,
fioipcov p£ fifi' p" — 47° 42' 40". Similarly, rpr^pdrcov ££
8' vz" — 67 P 4' 55", where p denotes the segment (of the
diameter). Where there was no unit, or no number of
sixtieths* second sixtieths, &c, the symbol O, signifying
ovSepicc fioTpa, ov8ev e^-qKocrrov, and the like, was used ; thus
poipvv 6 a ft" O"' = 0°1' 2 // /// . The system is parallel to
our system of decimal fractions, with the difference that the
submultiple is ■£$ instead of -£§ ; nor is it much less easy to
work with, while it furnishes a very speedy way of approxi-
mating to the values of quantities not expressible in whole
numbers. For example, in his Table of Chords, Ptolemy says
that the chord subtending an angle of 120° at the centre is
(r prf pdrcov) py ve' Ky" or 103P 55' 23"; this is equivalent
(since the radius of the circle is 60 rprjpaTa) to saying that
43 55 23
V3 = 1 +— + — - 9 + ---,, and this works out to 1-7320509...,
60 60 2 60"
which is correct to the seventh decimal place, and exceeds
the true value by 0-00000003 only.
46 GREEK NUMERICAL NOTATION
Practical calculation.
(a) The abacus.
In practical calculation it was open to the Greeks to secure
the advantages of a position-value system by using the
abacus. The essence of the abacus was the arrangement of
it in columns which might be vertical or horizontal, but were
generally vertical, and pretty certainly so in Greece and
Egypt; the columns were marked off by lines or in some
other way and allocated to the successive denominations of
the numerical system in use, i.e., in the case of the decimal
system, the units, tens, hundreds, thousands, myriads, and so
on. The number of units of each denomination was shown in
each column by means of pebbles, pegs, or the like. When,
in the process of addition or multiplication, the number of
pebbles collected in one column becomes sufficient to make
one or more units of the next higher denomination, the num-
ber of pebbles representing the complete number of the higher
units is withdrawn from the column in question and the
proper number of the higher units added to the next higher
column. Similarly, in subtraction, when a number of units of
one denomination has to be subtracted and there are not
enough pebbles in the particular column to subtract from, one
pebble from the next higher column is withdrawn and actually
or mentally resolved into the number of the lower units
equivalent in value ; the latter number of additional pebbles
increases the number already in the column to a number from
which the number to be subtracted can actually be withdrawn.
The details of the columns of the Greek abacus have unfor-
tunately to be inferred from the corresponding details of the
Roman abacus, for the only abaci which have been preserved
and can with certainty be identified as such are Roman.
There were two kinds; in one of these the marks were
buttons or knobs which could be moved up and down in each
column, but could not be taken out of it, while in the other
kind they were pebbles which could also be moved from one
column to another. Each column was in two parts, a shorter
portion at the top containing one button only, which itself
represented half the number of units necessary to make up
one of the next higher units, and a longer portion below
PRACTICAL CALCULATION 47
containing one less than half the same number. This arrange-
ment of the columns in two parts enabled the total number of
buttons to be economized. The columns represented, so far as
integral numbers were concerned, units, tens, hundreds, thou-
sands, &c, and in these cases the one button in the top
portion of each column represented five units, and there were
four buttons in the lower portion representing four units.
But after the columns representing integers came columns
representing fractions ; the first contained buttons represent-
ing unciae, of which there were 12 to the unit, i.e. fractions
of x^th, and in this case the one button in the top portion
represented 6 unciae or T \ths, while there were 5 buttons in
the lower portion (instead of 4), the buttons in the column
thus representing in all 1 1 unciae or 1 2ths. After this column
there were (in one specimen) three other shorter ones along-
side the lower portions only of the columns for integers, the
first representing fractions of aV^h (one button), the second
fractions of ^gth (one button), and the third fractions of y^nd
(two buttons, which of course together made up ^th).
The mediaeval writer of the so-called geometry of Boetius
describes another method of indicating in the various columns
the number of units of each denomination. 1 According to him
' abacus ' was a later name for what was previously called
mensa Pythagorea, in honour of the Master who had taught
its use. The method was to put in the columns, not the neces-
sary number of pebbles or buttons, but the corresponding
numeral, which might be written in sand spread over the
surface (in the same way as Greek geometers are said to have
drawn geometrical figures in sand strewn on boards similarly
called d/3a£ or afiaKiov). The figures put in the columns were
called apices. The first variety of numerals mentioned by the
writer are rough forms of the Indian figures (a fact which
proves the late date of the composition) ; but other forms were
(1) the first letters of the alphabet (which presumably mean
the Greek alphabetic numerals) or (2) the ordinary Roman
figures.
We should expect the arrangement of the Greek abacus to
correspond to the Roman, but the actual evidence regarding its
form and the extent to which it was used is so scanty that
1 Boetius, De Inst. Ar,, eel. Friedlein, pp. 396 sq.
48 GREEK NUMERICAL NOTATION
we may well doubt whether any great use was made of it at
all. But the use of pebbles to reckon with is attested by
several writers. In Aristophanes (Wasps, 656-64) Bdelycleon
tells his father to do an easy sum ' not with pebbles but with
fingers ', as much as to say, ' There is no need to use pebbles
for this sum ; you can do it on your fingers.' ' The income
of the state ', he says, 'is 2000 talents ; the yearly payment
to the 6000 dicasts is only 150 talents.' ' Why ', answers the
old man, ' we don't get a tenth of the revenue.' The calcula-
tion in this case amounted to multiplying 150 by 10 to show
that the product is less than 2000. But more to the purpose
are the following allusions. Herodotus says that, in reckoning
with pebbles, as in writing, the Greeks move their hand from
left to right, the Egyptians from right to left x ; this indicates
that the columns were vertical, facing the reckoner. Diogenes
Laertius attributes to Solon a statement that those who had
influence with tyrants were like the pebbles on a reckoning-
board, because they sometimes stood for more and sometimes
for less. 2 A character in a fourth-century comedy asks for an
abacus and pebbles to do his accounts. 3 But most definite of
all is a remark of Polybius that ' These men are really like
the pebbles on reckoning-boards. For the latter, according
to the pleasure of the reckoner, have the value, now of a
XclXkovs (Jth of an obol or ^th of a drachma), and the next
moment of a talent.' 4 The passages of Diogenes Laertius and
Polybius both indicate that the pebbles were not fixed in the
columns, but could be transferred from one to another, and
the latter passage has some significance in relation to the
Salaminian table presently to be mentioned, because the talent
and the x a ^ K °vs are actually the extreme denominations on
one side of the table.
Two relics other than the Salaminian table may throw
some light on the subject. First, the so-called Darius-vase
found at Canosa (Canusium), south-west of Barletta, represents
a collector of tribute of distressful countenance with a table in
front of him having pebbles, or (as some maintain) coins, upon
it and, on the right-hand edge, beginning on the side farthest
away and written in the direction towards him, the letters
1 Herodotus, ii. c. 36. 2 Diog. L. i. 59.
;> Alexis in Athenaeus, 117 c. 4 Polybius, v. 26. 13. •
PRACTICAL CALCULATION 49
MH'H >ro<T, while in his left hand he holds a sort of book in
which, presumably, he has to enter the receipts. Now M, 'f
(= X), H, and > are of course the initial letters of the words
for 10000, 1000, 100, and 10 respectively. Here therefore we
have a purely decimal system, without the halfway numbers
represented by P (= 7wre, 5) in combination with the other
initial letters which we find in the ' Attic ' s}^stem. The sign
P after > seems to be wrongly written for P, the older sign
for a drachma, O stands for the obol, < for the -|-obol, and T
{TeraprrjfMopioy) for the J-obol. 1 Except that the fractions of
the unit (here the drachma) are different from the fractions
of the Roman unit, this scheme corresponds to the Roman,
and so far might represent the abacus. Indeed, the decimal
arrangement corresponds better to the abacus than does the
Salaminian table with its intermediate ' Herodianic ' signs for
500, 50, and 5 drachmas. Prof. David Eugene Smith is, how-
ever, clear that any one can se,e from a critical examination of
the piece that what is represented is an ordinary money-
changer or tax-receiver with coins on a table such as one
might see anywhere in the East to-day, and that the table has
no resemblance to an abacus. 2 On the other hand, it is to be
observed that the open book held by the tax-receiver in his
left hand has TAAN on one page and TA1H on the other,
which would seem to indicate that he was entering totals in
talents and must therefore presumably have been adding coins
or pebbles on the table before him.
There is a second existing monument of the same sort,
namely a so-called ar/Kccpa (or arrangement of measures)
discovered about forty years ago 3 ; it is a stone tablet with
fluid measures and has, on the right-hand side, the numerals
XFHFAPKTIC. The signs are the 'Herodianic', and they
include those for 500, 50, and 5 drachmas ; h is the sign for
a drachma, T evidently stands for some number of obols
making a fraction of the drachma, i.e. the TpicofioXov or 3
obols, I for an obol, and C for a ^-obol.
The famous Salaminian table was discovered by Rangabe,
who gave a drawing and description of it immediately after-
1 Keil in Hermes, 29 r 1894, pp. 262-3.
2 Bibliotheca Mathematica, ix 3 , p. 193.
:1 Dumont in Revue arch eologi que, xxvi (1873), p. 43.
1523 Yj
50
GREEK NUMERICAL NOTATION
wards (1846). 1 The table, now broken into two unequal parts,
is in the Epigraphical Museum at Athens. The facts with
regard to it are stated, and a photograph of it is satisfactorily
produced, by Wilhelm Kubitschek. 2 A representation of it is
also given by Nagl 3 based on Rangabe^s description, and the
sketch of it here appended follows Nagl's drawing. The size
and material of the table (according to Rangabe"s measure-
ments it is 1-5 metres long and 0«75 metre broad) show that
it was no ordinary abacus ; it may
have been a fixture intended for
quasi-public use, such as a banker's
or money-changer's table, or again
it may have been a scoring-table
for some kind of game like tric-
trac or backgammon. Opinion has
from the first been divided between
the two views ; it has even been
suggested that the table was in-
tended for both purposes. But there
can be no doubt that it was used
for some kind of calculation and,
if it was not actually an abacus, it
may at least serve to give an idea
of what the abacus was like. The
difficulties connected with its in-
terpretation are easily seen. The
series of letters on the three sides are the same except
that two of them go no higher than X (1000 drachmae),
but the third has P (5000 drachmae), and T (the talent or
6000 drachmae) in addition; h is the sign for a drachma,
I for an obol (Jth of the drachma), C for -J-obol, T for ^-obol
(Teraprrj/jLopLoi/, Boeckh's suggestion), not J-obol (TpiTrj/jLopiov,
Vincent), and X for |-obol (x^kovs). It seems to be
agreed that the four spaces provided between the five shorter
lines were intended for the fractions of the drachma ; the first
space would require 5 pebbles (one less than the 6 obols
making up a drachma), the others one each. The longer
1 Revue archeologique, iii. 1846.
2 Wiener numismatische Zeitschrift, xxxi. 1899, pp. 393-8, with
Plate xxiv.
3 Abh. zur Gesch. d. Math. ix. 1899, plate after p. 357.
X^HPAri-ICTX
%
v
-1
X X
X o
1
> b\
~* $
o I
H E.
PRACTICAL CALCULATION 51
lines would provide the spaces for the drachmae and higher
denominations. On the assumption that the cross line indi-
cates the Roman method of having one pebble above it to
represent 5, and four below it representing units, it is clear
that, including denominations up to the talent (6000 drachmae),
only five columns are necessary, namely one for the talent or
6000 drachmae, and four for 1000, 100, 10 drachmae, and 1
drachma respectively. But there are actually ten spaces pro-
vided by the eleven lines. On the theory of the game-board,
five of the ten on one side (right or left) are supposed to
belong to each of two players placed facing each other on the
two longer sides of the table (but, if in playing they had to
use the shorter columns for the fractions, it is not clear how
they would make them suffice) ; the cross on the middle of the
middle line might in that case serve to mark the separation
between the lines belonging to the two players, or perhaps all
the crosses may have the one object of helping the eye to dis-
tinguish all the columns from one another. On the assump-
tion that the table is an abacus, a possible explanation of the
eleven lines is to suppose that they really supply five columns
only, the odd lines marking the divisions between the columns,
and the even lines, one in the middle of each column,
marking where the pebbles should be placed in rows ; in this
case, if the crosses are intended to mark divisions between the
four pebbles representing units and the one pebble represent-
ing 5 in each column, the crosses are only required in the last
three columns (for 100, 10, and 1), because, the highest de-
nomination being 6000 drachmae, there was no need for a
division of the 1000-column, which only required five unit-
pebbles altogether. Nagl, a thorough-going supporter of the
abacus-theory to the exclusion of the other, goes further and
shows how the Salaminian table could have been used for the
special purpose of carrying out a long multiplication ; but this
development seems far-fetched, and there is no evidence of
such a use.
The Greeks in fact had little need of the abacus for calcu-
lations. With their alphabetic numerals they could work out
their additions, subtractions, multiplications, and divisions
without the help of any marked columns, in a form little less
convenient than ours : examples of long multiplications, which
e 2
52 GREEK NUMERICAL NOTATION
include addition as the last step in each case, are found in
Eutocius's commentary on Archimedes's Measurement of
a Circle. We will take the four arithmetical operations
separately.
,(/3) Addition and Subtraction.
There is no doubt that, in writing down numbers for the
purpose of these operations, the Greeks would keep the several
powers of 10 separate in a manner practically corresponding
to our system of numerals, the hundreds, thousands, &c., being
written in separate vertical rows. The following would be
a typical example of a sum in addition :
/ a v k 8 -
1424
P 7
103
M ficnra
12281
M A
30030
8
M / ya)A?7
43838
and the mental part of the work would be the same for the
Greek as for us.
Similarly a subtraction would be represented as follows :
e
M/yxA^- = 93636
M y yv 23409
M <tk( 70227
(y) Multiplication.
(i) The Egyptian method.
For carrying out multiplications two things were required.
The first was a multiplication table. This the Greeks are
certain to have had from very early times. The Egyptians,
indeed, seem never to have had such a table. We know from
the Papyrus Rhind that in order to multiply by any number
the Egyptians began by successive doubling, thus obtaining
twice, four times, eight times, sixteen times the multiplicand,
and so on ; they then added such sums of this series of multi-
ples (including once the multiplicand) as were required. Thus,
MULTIPLICATION 53
to multiply by 13, they did not take 10 times and 3 times
the multiplicand respectively and add them, but they found
13 times the multiplicand by adding once and 4 times and 8
times it, which elements they had obtained by the doubling
process; similarly they would find 25 times any number by
adding once and 8 times and 16 times the number. 1 Division
was performed by the Egyptians in an even more rudimen-
tary fashion, namely by a tentative back-multiplication begin-
ning with the same doubling process. But, as we have seen
(p. 14), the scholiast to the Charmides says that the branches
of Xoyio-TiKrj include the ' so-called Greek and Egyptian
methods in multiplications and divisions'.
(ii) The Greek method.
The Egyptian method being what we have just described, it
seems clear that the Greek method, which was different,
depended on the direct use of a multiplication table. A frag-
ment of such a multiplication table is preserved on a two-
leaved wax tablet in the British Museum (Add. MS. 34186).
1 I have been told that there is a method in use to-day (some say in
Russia, but I have not been able to verify this), which is certainly attractive
and looks original, but which will immediately be seen to amount simply
to an elegant practical method of carrying out the Egyptian procedure.
Write out side by side in successive lines, so as to form two columns,
(1) the multiplier and multiplicand, (2) half the multiplier (or the
nearest integer below it if the multiplier is odd) and twice the multi-
plicand, (3) half (or the nearest integer below the half) of the number
in the first column of the preceding row and twice the number in the
second column of the preceding row, and so on, until we have 1 in
the first column. Then strike out all numbers in the second column
which are opposite even numbers in the first column, and add all the
numbers left in the second column. The sum will be the required
product. Suppose e.g. that 157 is to be multiplied by 83. The rows
and columns then are :
83 157
41 314
20 -628-
10 1256-
5 2512
2 -502*
1 10048
13031 = 83x157
The explanation is, of course, that, where we take half the preceding
number in the first column less one, we omit once the figure in the right-
hand column, so that it must be left in that column to be added in at
the end; and where we take the exact half of an even number, we
omit nothing in the right-hand column, but the new line is the exact
equivalent of the preceding one, which can therefore be struck out.
54 GREEK NUMERICAL NOTATION
It is believed to date from the second century a.d., and it
probably came from Alexandria or the vicinity. But the
form of the characters and the mingling of capitals and small
letters both allow of an earlier date ; e.g. there is in the
Museum a Greek papyrus assigned to the third century B.C.
in which the numerals are very similar to those on the tablet. 1
The second requirement is connected with the fact that the
Greeks began their multiplications by taking the product of
the highest constituents first, i.e. they proceeded as we should
if we were to begin our long multiplications from the left
instead of the right. The only difficulty would be to settle
the denomination of the products of two high powers of ten.
With such numbers as the Greeks usually had to multiply
there would be no trouble ; but if, say, the factors were un-
usually large numbers, e.g. millions multiplied by millions or
billions, care would be required, and even some rule for
settling the denomination, or determining the particular
power or powers of 10 which the product would contain.
This exceptional necessity was dealt with in the two special
treatises, by Archimedes and Apollonius respectively, already
mentioned. The former, the Sand-reckoner, proves that, if
there be a series of numbers, 1, 10, 10 2 , 10 3 ... 10 m ... 10 w ...,
then, if 10 m , 10 w be any two terms of the series, their product
10™ . 10 n will be a term in the same series and will be as many
terms distant from 10 n as the term \0 m is distant from 1 ;
also it will be distant from 1 by a number of terms less by
one than the sum of the numbers of terms by which 1 m and
10 n respectively are distant from 1. This is easily seen to be
equivalent to the fact that, 1 m being the (m + 1 )th term
beginning with 1, and 10 w the (?i+l)th term beginning
with 1, the product of the two terms is the (m + ^+l)th
term beginning with 1, and is 10 m+w .
(iii) Apollonius's continued multiplications.
The system of Apollonius deserves a short description. 2 Its
object is to give a handy method of finding the continued
product of any number of factors, each of which is represented
by a single letter in the Greek numeral notation. It does not
1 David Eugene Smith in Bibliotheca Mathematica, ix 3 , pp. 193-5.
2 Our authority here is the Synagoge of Pappus, Book ii, pp. 2-28, Hultsch.
MULTIPLICATION 55
therefore show how to multiply two large numbers each of
which contains a number of digits (in our notation), that is,
a certain number of units, a certain number of tens, a certain
number of hundreds, &c. ; it is confined to the multiplication
of any number of factors each of which is one or other of the
following : (a) a number of units as 1, 2, 3, ... 9, (b) a number
of even tens as 10, 20, 30, ... 90, (c) a number of even hundreds
as 100, 200, 300, ... 900. It does not deal with factors above
hundreds, e.g. 1000 or 4000; this is because the Greek
numeral alphabet only went up to 900, the notation begin-
ning again after that with a, ,/3, . . . for 1000, 2000, &c. The
essence of the method is the separate multiplication (1) of the
bases, irvQ\ikvv$, of the several factors, (2) of the powers of ten
contained in the factors, that is, what we represent by the
ciphers in each factor. Given a multiple of ten, say 30, 3 is
the TTvOjjirjv or base, being the same number of units as the
number contains tens ; similarly in a multiple of 100, say 800,
8 is the base. In multiplying three numbers such as 2, 30,
800, therefore, Apollonius first multiplies the bases, 2, 3, and 8,
then finds separately the product of the ten and the hundred,
and lastly multiplies the two products. The final product has
to be expressed as a certain number of units less than a
myriad, then a certain number of myriads, a certain number
of ' double myriads ' (myriads squared), ' triple myriads '
(myriads cubed), &c, in other words in the form
A + A 1 M+A 2 M* + ... ,
where M is a myriad or 10 4 and A , A x ... respectively repre-
sent some number not exceeding 9999.
No special directions are given for carrying out the multi-
plication \ of the bases (digits), or for the multiplication of
their product into the product of the tens, hundreds, &c,
when separately found (directions for the latter multiplica-
tion may have been contained in propositions missing from
the mutilated fragment in Pappus). But the method of deal-
ing with the tens and hundreds (the ciphers in our notation)
is made the subject of a considerable number of separate
propositions. Thus in two propositions the factors are all of
one sort (tens or hundreds), in another we have factors of two
sorts (a number of factors containing units only multiplied
56 GREEK NUMERICAL NOTATION
by a number of multiples of ten, each less than 100, or by
multiples of 100, each less than 1000), and so on. In the final
proposition (25), with which the introductory lemmas close,
the factors are of all three kinds, some containing units only,
others being multiples of 10 (less than 100) and a third set
being multiples of 100 (less than 1000 in each case). As
Pappus frequently says, the proof is easy ' in numbers ' ;
Apollonius himself seems to have proved the propositions by
means of lines or a diagram in some form. The method is the
equivalent of taking the indices of all the separate powers of
ten included in the factors (in which process ten = 10* counts
as 1. and 100 = 10 2 as 2), adding the indices together, and then
dividing the sum by 4 to obtain the power of the myriad
(10000) which the product contains. If the whole number in
the quotient is n, the product contains (10000) w or the
n -myriad in Apollonius's notation. There will in most cases
be a remainder left after division by 4, namely 3, 2, or 1 : the
remainder then represents (in our notation) 3, 2, or 1 more
ciphers, that is, the product is 1000, 100, or 10 times the
^-myriad, or the 10000^, as the case may be.
We cannot do better than illustrate by the main problem
which Apollonius sets himself, namely that of multiplying
together all the numbers represented by the separate letters
in the hexameter :
'Apri/JLiSos K\etT€ Kpdros €^o\ov kvvia Kovpai.
The number of letters, and therefore of factors, is 38, of which
10 are multiples of 100 less than 1000, namely p, r, cr, r, p, r,
a-, x , v, p (=100, 300, 200, 300, 100, 300, 200, 600, 400, 100),
1 7 are multiples of 10 less than 100, namely //, i, o, k, A, i, k, o, £,
o, o, v, i>, v, k, o, l ( = 40, 10, 70, 20, 30, 10, 20, 70, 60, 70, 70, 50,
50, 50, 20, 70, 10), and 11 are numbers of units not exceeding
9, namely a, e, 8, e, e, a, e, e, e, a, a ( = 1, 5, 4, 5, 5, 1, 5, 5, 5, 1, 1).
The sum of the indices of powers of ten contained in the
factors is therefore 10.2 + 17.1=37. This, when divided by
4, gives 9 with 1 as remainder. Hence the product of all the
tens and hundreds, excluding the bases in each, is 10 . 10000 9 .
We have now, as the second part of the operation, to mul-
tiply the numbers containing units only by the bases of all the
other factors, i.e. (beginning with the bases, first of the hun-
dreds, then of the tens) to multiply together the numbers :
MULTIPLICATION
57
1, 3, 2, 3, 1, 3, 2,6,4, 1,
4, 1, 7, 2, 3, 1, 2, 7, 6, 7, 7, 5, 5, 5, 2, 7, 1,
and 1, 5, 4, 5, 5, 1, 5, 5, 5, 1, 1.
The product is at once given in the text as 1 9 ' quadruple
myriads ', 6036 ' triple myriads ', and 8480 ' double myriads ', or
19. 10000 4 + 6036. 10000 3 + 8480. 10000 2 .
(The detailed multiplication line by line, which is of course
perfectly easy, is bracketed by Hultsch as interpolated.)
Lastly, says Pappus, this product multiplied by the other
(the product of the tens and hundreds without the bases),
namely 10 . 10000 9 , as above, gives
196. 10000 13 + 368. 10000 12 + 4800 . 10000 11 .
(iv) Examples of ordinary multiplications.
I shall now illustrate, by examples taken from Eutocius, the
Greek method of performing long multiplications. It will be
seen that, as in the case of addition and subtraction, the
working is essentially the same as ours. The multiplicand is
written first, and below it is placed the multiplier preceded by
km (= ' by ' or ' into '). Then the term containing the highest
power of 10 in the multiplier is taken and multiplied into all
the terms in the multiplicand, one after the other, first into that
containing the highest power of 10, then into that containing
the next highest power of 10, and so on in descending order ;
after which the term containing the next highest power of 10
in the multiplier is multiplied into all the terms of the multi-
plicand in the same order ; and so on. The same procedure
is followed where either or both of the numbers to be multi-
plied contain fractions. Two examples from Eutocius will
make the whole operation clear.
(1) jOLTVCL
kiri arva
1351
1351
MMM/i
A a
MMM.er
1000000
300000
300000 50000
90000 15000
50000 15000
1000
1000
300
2500 50
300 50
o/jLov M / ecra
together 1825201
58 GREEK NUMERICAL NOTATION
(2) t yiyV$ 3013|i [=3013|]
km ,yiyV$ x 3013JJ
9000000
30000
9000
1500
750
M.pXe/31'
30000
100
30
5
2i
^2
fiXOal'l'V
9000
30
9
H
1 1
2 4
■tyv $W \s' 8 7)' l<?'
1500
750
5
^2
i i
2 4
i
4
1
8
1
8
1
Te
6/iov MfixTrO nj-' together 9082689 T V
The following is one among many instances in which Heron
works out a multiplication of two numbers involving fractions.
He has to multiply 4§§ by 7§f , which he effects as follows
(Geom. 12. 68):
4.7 --= 28,
A 62 _2_48
^ • 64 — 64'
.3 3 7 2 31
64 ' ' ~ 6~4~
3.3 62 2046 _1_ 3 1 i 62 JL_ •
64*64 — 64 * 64 — 64 T 64' 64?
the result is therefore
9 ft 510 i 6 2 1 O O i 7 j3 2 i j5 2^ 1
^° R A I fi A • fi A ""T ' ft A T ft A • Sd
64 ' 64'64 — " w ' * 64 ' 64*64
— *}*> 62 .62 JL_
— °° 64 ' 64 * 64*
The multiplication of 37° 4' 55" (in the sexagesimal system)
by itself is performed by Theon of Alexandria in his com-
mentary on Ptolemy's Syntaxis in an exactly similar manner.
(S) Division.
The operation of division depends on those of multiplication
and subtraction, and was performed by the Greeks, mutatis
mutandis, in the same way as we perform it to-day. Suppose,
for example, that the process in the first of the above multi-
P TT/3
plications had to be reversed and M/o-a (1825201) had to be
divided by ,cltvcl (1351). The terms involving the successive
powers of 10 would be mentally kept separate, as in addition
and subtraction, and the first question would be, how many
times does one thousand go into one million, allowing for the
fact that the one thousand has 351 behind it, while the one
million has 825 thousands behind it. The answer is one
thousand or 7 a, and this multiplied by the divisor t arva gives
pAe pT/3 H<T
M y a which, subtracted from M^cra, leaves Mfoa. This
DIVISION
59
remainder ( = 474201) has now to be divided by ^arva (1351),
and it would be seen that the latter would go into the former
r (300) times, but not v (400) times. Multiplying aurva by r,
we obtain M^r (405300), which, when subtracted from Mfoa
(474201), leaves M^^a (68901). This has again to be divided
by s aTva and goes v (50) times ; multiplying t arva by v, we
have M/0z/ (67550), which, subtracted from Mrf^a (68901),
leaves y aTua (1351). The last quotient is therefore a (1), and
the whole quotient is y arua (1351).
An actual case of long division where both dividend and
divisor contain sexagesimal fractions is described by Theon.
The problem is to divide 1515 20' 15" by 25 12' 10", and
Theon's account of the process amounts to the following :
Divisor. Dividend.
25 12' 10" 1515 20' 15'
25. 60 = 1500
Quotient.
First term 60
Remainder 15:
= 900'
Sum
12'. 60 =
920'
720'
Remainder
10". 60 =
200'
10'
Remainder
25.7' =
190'
175'
15' =
= 900"
Sum
12'. 7' :
915"
84"
Rema
10". 7':
inder
binder
831"
1" 10'"
Rema
25.33"
829" 50'"
825"
Second term 7'
Third
term 33"'
Remainder
12'. 33" =
4" 50'
290'
396'
(too great by) 106"
Thus the quotient is something less than 60 7' 33". It will
be observed that the difference between this operation of
60 GREEK NUMERICAL NOTATION
pT/3
Theon's and that of dividing M^cra by arva as above is that
Theon makes three subtractions for one term of the quotient,
whereas the remainder was arrived at in the other case after
one subtraction. The result is that, though Theon's method
is quite clear, it is longer, and moreover makes it less easy to
foresee what will be the proper figure to try in the quotient,
so that more time would probably be lost in making un-
successful trials.
(e) Extraction of the square root.
We are now in a position to see how the problem of extract-
ing the square root of a number would be attacked. First, as
in the case of division, the given whole number would be
separated into terms containing respectively such and such
a number of units and of the separate powers of 10. Thus
there would be so many units, so many tens, so many hun-
dreds, &c, and it would have to be borne in mind that the
squares of numbers from 1 to 9 lie between 1 and 99, the
squares of numbers from 10 to 90 between 100 and 9900, and
so on. Then the first term of the square root would be some
number of tens or hundreds or thousands, and so on, and
would have to be found in much the same way as the first
term of a quotient in a long division, by trial if necessary.
If A is the number the square root of which is required, while
a represents the first term or denomination of the square root,
and x the next term or denomination to be found, it would be
necessary to use the identity (a + x) 2 = a 2 + 2ax + x 2 and to
find x so that 2ax + x 2 might be somewhat less than the
remainder A —a 2 , i.e. we have to divide A — a 2 by 2a, allowing
for the fact that not only must 2 ax (where x is the quotient)
but also (2a + x)x be less than A— a 2 . Thus, by trial, the
highest possible value of x satisfying the condition would be
easily found. If that value were b, the further quantity
2 ab + b 2 would have to be subtracted from the first remainder
A — a 2 , and from the second remainder thus left a third term
or denomination of the square root would have to be found in
like manner; and so on. That this was the actual procedure
followed is clear from a simple case given by Theon of Alex-
andria in his commentary on the Syntaxis. Here the square
root of 144 is in question, and it is obtained by means of
EXTRACTION OF THE SQUARE ROOT
61
Eucl. II. 4. The highest possible denomination (i.e. power
of 10) in the square root is 10 ; 10 2 subtracted from 144 leaves
44, and this must contain, not only twice the product of 10
and the next term of the square root, but also the square of
the next term itself. Now twice 1.10 itself produces 20, and
the division of 44 by 20 suggests 2 as the next term of the
square root ; this turns out to be the exact figure required, since
2.20 + 2 2 = 44.
The same procedure is illustrated by Theon's explanation
of Ptolemy's method of extracting square roots according to
the sexagesimal system of fractions. The problem is to find
approximately the square root of 4500 fioipoa or degrees, and
K
67
H
67°
4489
F
4'
268'
55"
b
00
00
CD
CO
4-' 268'
16
55" 3688" 40 "'
L
a geometrical figure is used which proves beyond doubt the
essentially Euclidean basis of the whole method. The follow-
ing arithmetical representation of the purport of the passage,
when looked at in the light of the figure, will make the
matter clear. Ptolemy has first found the integral part of
7(4500) to be 67. Now 67 2 = 4489, so that the remainder is
1 1 . Suppose now that the rest of the square root is expressed
by means of sexagesimal fractions, and that we may therefore
write
7(4500)= 6 7 + — + -^- ,
v ; 60 60 2
where x, y are yet to be found. Thus x must be such that
2 . 67x/60 is somewhat less than 11, or x must be somewhat
62 GREEK NUMERICAL NOTATION
less than — - — or - , which is at the same time greater than
2.67 67 ' s
4. On trial it turns out that 4 will satisfy the conditions of
/ 4 \ 2
the problem, namely that ( 67 + -- ) must be less than 4500,
so that a remainder will be left by means of which y can be
found.
2.67.4 / 4 \ 2
Now this remainder is 1 1 — ( — ) j and this is
60 ^60/
equal to 11 . 60 2 -2 . 67 . 4 . 60-16 7424
~602~ 01 * ~t¥"
Thus we must suppose that 2 (67 -\ j -~ approximates to
7424
- ^z > or that 8048^/ is approximately equal to 7424.60.
Therefore y is approximately equal to 55.
/ 4 \ 55 /55 \ 2
We have then to subtract 2(67H ) — „ + (^^) » or
V 60/ 60 2 \60 2 /
442640 3025 „ .. . , 7424 . , 3
—^i — H -zr^r ' irom the remainder - above found.
60" 60 4 60 2
™ ,, ,. „ 442640, 7424 . 2800 46 40
The subtraction of -^- from _ gives -^ or — + — 3 ;
but Theon does not go further and subtract the remaining
—-4- ; he merely remarks that the square of -— 2 approximates
to — ^ + —To. As a matter of fact, if we deduct the „. ■ from
60 2 60 3 60 4
■ — — , so as to obtain the correct remainder, it is found
60* '
, 164975
tobe "60\-
Theon' s plan does not work conveniently, so far as the
determination of the first fractional term (the first-sixtieths)
is concerned, unless the integral term in the square root is
9
X / 0C \"
large relatively to— - ; if this is not the case, the term ( -*— j is
not comparatively negligible, and the tentative ascertainment
of x is more difficult. Take the case of Vs, the value of which,
43 55 23
in Ptolemy's Table of Chords, is equal to 1 -\ 1 n -\ »•
J ' ^ 60 60 2 60- 3
EXTRACTION OF THE SQUARE ROOT 63
If we first found the unit 1 and then tried to find the next
term by trial, it would probably involve a troublesome amount
of trials. An alternative method in such a case was to
multiply the number by 60 2 , thus reducing it to second-
sixtieths, and then, taking the square root, to ascertain the
number of first-sixtieths in it. Now 3.60 2 = 10800, and, as
103 2 = 10609, the first element in the square root of 3 is
found in this way to be — — - ( = 1 + — ). That this was the
method in such cases is indicated by the fact that, in the Table
of Chords, each chord is expressed as a certain number of
first-sixtieths, followed by the second-sixtieths, &c, \/3 being
expressed as \- — 5 H « • The same thing; is indicated by
1 60 60 2 60 3 s J
the scholiast to Eucl., Book X, w T ho begins the operation of
finding the square root of 31 10' 36" by reducing this to
second-sixtieths; the number of second-sixtieths is 112236,
which gives, as the number of first-sixtieths in the square
335
root, 335, while — - = 5 35'. The second-sixtieths in the
60
square root can then be found in the same way as in Theon's
example. Or, as the scholiast says, we can obtain the square
root as far as the second-sixtieths by reducing the original
number to fourth-sixtieths, and so on. This would no doubt
be the way in which the approximate value 2 49' 42" 20'" 10""
given by the scholiast for \/8 was obtained, and similarly
with other approximations of his, such as V2 =1 24' 51" and
V(27) =5 11' 46" 50'" (the 50 w should be 10"').
(^) Extraction of the cube root
Our method of extracting the cube root of a number depends
upon the formula (a + x) z = a 3 + 3a 2 x -f 3 ax 2 + a? 3 , just as the
extraction of the square root depends on the formula
(a + x) 2 = a 2 + 2ax-\-x 2 . As we have seen, the Greek method
of extracting the square root was to use the latter (Euclidean)
formula just as we do ; but in no extant Greek writer do we
find any description of the operation of extracting the cube
root. It is possible that the Greeks had not much occasion
for extracting cube roots, or that a table of cubes would
suffice for most of their purposes. But that they had some
64 GREEK NUMERICAL NOTATION
9
method is clear from a passage of Heron, where he gives 4 —
as an approximation to JV(IOO), and shows how he obtains it. 1
Heron merely gives the working dogmatically, in concrete
numbers, without explaining its theoretical basis, and we
cannot be quite certain as to the precise formula underlying
the operation. The best suggestion which has been made on
the subject will be given in its proper place, the chapter
on Heron.
1 Heron, Metrica, iii. c. 20.
Ill
PYTHAGOREAN ARITHMETIC
There is very little early evidence regarding Pythagoras's
own achievements, and what there is does not touch his mathe-
matics. The earliest philosophers and historians who refer
to him would not be interested in this part of his work.
Heraclitus speaks of his wide knowledge, but with disparage-
ment : ' much learning does not teach wisdom ; otherwise
it would have taught Hesiod and Pythagoras, and again
Xenophanes and Hecataeus '} Herodotus alludes to Pytha-
goras and the Pythagoreans several times ; he calls Pythagoras
' the .most able philosopher among the Greeks ' (EXXrjucou ov
tco da-devea-rccTCo (to^lo-ttj IlvOayopr]). 2 In Empedocles he had
an enthusiastic admirer : ' But there was among them a man
of prodigious knowledge who acquired the profoundest wealth
of understanding and was the greatest master of skilled arts
of every kind ; for, whenever he willed with his whole heart,
he could with ease discern each and every truth in his ten —
nay, twenty — men's lives.' 3
Pythagoras himself left no written exposition of his
doctrines, nor did any of his immediate successors, not even
Hippasus, about whom the different stories ran (1) that he
was expelled from the school because he published doctrines
of Pythagoras, and (2) that he was drowned at sea for
revealing the construction of the dodecahedron in the sphere
and claiming it as his own, or (as others have it) for making
known the discovery of the irrational or incommensurable.
Nor is the absence of any written record of Pythagorean
1 Diog. L. ix. 1 (Fr. 40 in VorsoJcratiker, i 3 , p. 86. 1-3).
2 Herodotus, iv. 95.
3 Diog. L. viii. 54 and Porph. V. Pyth. 30 (Fr. 129 in Vors. i 3 , p. 272.
15-20).
1523 F
66 PYTHAGOREAN ARITHMETIC
doctrines clown to the time of Philolaus to be attributed
to a pledge of secrecy binding the school ; at all events, it
did not apply to their mathematics or their physics ; the
supposed secrecy may even have been invented to explain
the absence of documents. The fact appears to be that oral
communication was the tradition of the school, while their
doctrine would in the main be too abstruse to be understood
by the generality of people outside.
In these circumstances it is difficult to disentangle the
portions of the Pythagorean philosophy which can safely
be attributed to the founder of the school. Aristotle evi-
dently felt this difficulty ; it is clear that he knew nothing
for certain of any ethical or physical doctrines going back
to Pythagoras himself ; and when he speaks of the Pytha-
gorean system, he always refers it to ' the Pythagoreans ',
sometimes even to ' the so-called Pythagoreans '.
The earliest direct testimony to the eminence of Pythagoras
in mathematical studies seems to be that of Aristotle, who in
his separate book On the Pythagoreans, now lost, wrote that
' Pythagoras, the son of Mnesarchus, first worked at mathe-
matics and arithmetic, and afterwards, at one time, condescended
to the wonder-working practised by Pherecydes.' ]
In the Metaphysics he speaks in similar terms of the
Pythagoreans :
' In the time of these philosophers (Leucippus and
Democritus) and before them the so-called Pythagoreans
applied themselves to the study of mathematics, and were
the first to advance that science ; insomuch that, having been
brought up in it, they thought that its principles must be
the principles of all existing things.' 2
It is certain that the Theory of Numbers originated in
the school of Pythagoras ; and, with regard to Pythagoras
himself, we are told by Aristoxenus that he ' seems to have
attached supreme importance to the study of arithmetic,
which he advanced and took out of the region of commercial
utility \ 3
1 Apollonius, Hist, mirabil. 6 (Vors. i 3 , p. 29. 5).
2 Arist. Metaph. A. 5, 985 b 23.
3 Stobaeus, Ed. i. proem. 6 (Vors. i 3 , p. 346. 12).
PYTHAGOREAN ARITHMETIC 67
Numbers and the universe.
We know thatThales (about % 624-547 B.C.) and Anaximander
(born probably in 611/10 B.C.) occupied themselves with
astronomical phenomena, and, even before their time, the
principal constellations had been distinguished. Pythagoras
(about 572-497 B.C. or a little later) seems to have been
the first Greek to discover that the planets have an inde-
pendent movement of their own from west to east, i.e. in
a direction contrary to the daily rotation of the fixed stars ;
or he may have learnt what he knew of the planets from the
Babylonians. Now any one who was in the habit of intently
studying the heavens would naturally observe that each
constellation has two characteristics, the number of the stars
which compose it and the geometrical figure which they
form. Here, as a recent writer has remarked, 1 we find, if not
the origin, a striking illustration of the Pythagorean doctrine.
And, just as the constellations have a number characteristic
of them respectively, so all known objects have a number ;
as the formula of Philolaus states, ' all things which can
be known have number; for it is not possible that without
number anything can either be conceived or known '. 2
This formula, however, does not yet express all the content
of the Pythagorean doctrine. Not only do all things possess
numbers ; but, in addition, all things are numbers ; ' these
thinkers ', says Aristotle, ' seem to consider that number is
the principle both as matter for things and as constituting
their attributes and permanent states '. 3 True, Aristotle
seems to regard the theory as originally based on the analogy
between the properties of things and of numbers.
' They thought they found in numbers, more than in fire,
earth, or water, many resemblances to things which are and
become ; thus such and such an attribute of numbers is jus-
tice, another is soul and mind, another is opportunity, and so
on ; and again they saw in numbers the attributes and ratios
of the musical scales. Since, then, all other things seemed
in their whole nature to be assimilated to numbers, while
numbers seemed to be the first things in the whole of nature,
1 L. Brunschvicg-, Les etapes de la philosophie mathematique, 1912, p. 33.
2 Stob. Eel. i. 21, 7 b (Vors. i 3 , p. 310. 8-10).
3 Aristotle, Metaph. A. 5, 986 a 16.
F 2
68 PYTHAGOREAN ARITHMETIC
they supposed the elements of numbers to be the elements
of all things, and the whole heaven to be a musical scale and
a number.' l
This passage, with its assertion of ' resemblances ' and
' assimilation ', suggests numbers as affections, states, or rela-
tions rather than as substances, and the same is implied by
the remark that existing things exist by virtue of their
imitation of numbers. 2 But again we are told that the
numbers are not separable from the things, but that existing
things, even perceptible substances, are made up of numbers ;
that the substance of all things is number, that things are
numbers, that numbers are made up from the unit, and that the
whole heaven is numbers. 3 Still more definite is the statement
that the Pythagoreans ' construct the whole heaven out of
numbers, but not of monadic numbers, since they suppose the
units to have magnitude ', and that, ' as we have said before,
the Pythagoreans assume the numbers to have magnitude \ 4
Aristotle points out certain obvious difficulties. On the one
hand the Pythagoreans speak of ' this number of which the
heaven is composed ' ; on the other hand they speak of ' attri-
butes of numbers ' and of numbers as ' the causes of the things
which exist and take place in the heaven both from the begin-
ning and now '. Again, according to them, abstractions and
immaterial things are also numbers, and they place them in
different regions ; for example, in one region they place
opinion and opportunity, and in another, a little higher up or
lower down, such things as injustice, sifting, or mixing.
Is it this same 'number in the heaven' which we must
assume each of these things to be, or a number other than
this number 1 5
May we not infer from these scattered remarks of Aristotle
about the Pythagorean doctrine that 'the number in the
heaven ' is the number of the visible stars, made up of
units which are material points? And may this not be
the origin of the theory that all things are numbers, a
theory which of course would be confirmed when the further
1 Metaph. A. 5, 985 b 27-986 a 2. 2 lb. A. 5, 987 b 11.
3 lb. N. 3, 1090 a 22-23 ; M. 7, 1080 b 17 ; A. 5, 987 a 19, 987 b 27,
986 a 20.
* lb. M. 7, 1080 b 18, 32. 5 lb. A. 8, 990 a 18-29.
NUMBERS AND THE UNIVERSE 69
capital discovery was made that musical harmonies depend
on numerical ratios, the octave representing the ratio 2 : 1
in length of string, the fifth 3 : 2 and the fourth 4:3?
The use by the Pythagoreans of visible points to represent
the units of a number of a particular form is illustrated by
the remark of Aristotle that
' Eurytus settled what is the number of what object (e.g.
this is the number of a man, that of a horse) and imitated
the shapes of living things by pebbles after the manner of
those ivho bring numbers into the forms of triangle or
square \ l
They treated the unit, which is a point without position
{a-TLyfir) aOeros), as a point, and a point as a unit having
position (uovas Qkaiv e^ovcra). 2
Definitions of the unit and of number.
Aristotle observes that the One is reasonably regarded as
not being itself a number, because a measure is not the things
measured, but the measure or the One is the beginning (or
principle) of number. 3 This doctrine may be of Pythagorean
origin ; Nicomaehus has it 4 ; Euclid implies it when he says
that a unit is that by virtue of which each of existing things
is called one, while a number is ' the multitude made up of
units ' 5 ; and the statement was generally accepted. According
to Iamblichus, Thymaridas (an ancient Pythagorean, probably
not later than Plato's time) defined a unit as 'limiting quan-
tity' (irepaLvovcra iroa-oT-qs) or, as we might say, 'limit of few-
ness ', while some Pythagoreans called it ' the confine between
number and parts', i.e. that which separates multiples
and submultiples. Chrysippus (third century B.C.) called it
' multitude tone ' (nXrjOos eV), a definition objected to by
Iamblichus as a contradiction in terms, but important as an
attempt to bring 1 into the conception of number.
The first definition of number is attributed to Thales, who
defined it as a collection of units (jiovaScov o-varrjua), ' follow-
1 Metaph. N. 5, 1092 b 10.
2 lb. M. 8, 1084 b 25 ; De an. i. 4, 409 a 6 ; Proclus on Eucl. I, p. 95. 21.
3 Metaph.N. 1, 1088 a 6.
4 Nicora. Introd. arithm. ii. 6. 3, 7. 3. 5 Eucl. VII, Defs. 1, 2.
6 Iambi, in Nicom. ar. introd., p. 11. 2-10.
70 PYTHAGOREAN ARITHMETIC
ing the Egyptian view \ l The Pythagoreans ' made number
out of one' 2 ; some of them called it ' a progression of multi-
tude beginning from a unit and a regression ending in it \ 3
(Stobaeus credits Moderatus, a Neo-Pythagorean of the time
of Nero, with this definition. 4 ) Eudoxjis defined number as
a ' determinate multitude ' (nXrjOo? copier p.evov). b Nicoma-
chus has yet another definition, ' a flow of quantity made up
of units ' 6 (ttooSttitos X^f jLa * K J^opotScoy avyKeipevov). Aris-
totle gives a number of definitions equivalent to one or other
of those just mentioned, 'limited multitude', 7 'multitude (or
' combination ') of units ', 8 l multitude of indivisibles ', 9 ' several
ones' (eVa TrXeico), 10 ' multitude measurable by one', 11 'multi-
tude measured ', and ' multitude of measures ' 12 (the measure
being the unit).
Classification of numbers.
The distinction between odd (irepicrcros) and even (dpnos)
doubtless goes back to Pythagoras. A Philolaus fragment
says that ' number is of two special kinds, odd and even, with
a third, even-odd, arising from a mixture of the two ; and of
each kind there are many forms \ 13 According to Nicomachus,
the Pythagorean definitions of odd and even were these :
' An even number is that which admits of being divided, by
one and the same operation, into the greatest and the least
parts, greatest in size but least in number (i. e. into tivo halves)
. . ., while an odd number is that which cannot be so divided
but is only divisible into two unequal parts.' 14
Nicomachus gives another ancient definition to the effect
that
' an even number is that which can be divided both into two
equal parts and into two unequal parts (except the funda-
mental dyad which can only be divided into two equal parts),
but. however it is divided, must have its two parts of the same
hind without part in the other kind (i. e. the two parts are
1 Iambi, in Nicom. ar. introd., p. 10. 8-10.
2 Arist. Metaph. A. 5, 986 a 20. 3 Theon of Smyrna, p. 18. 3-5.
4 Stob. Eel. i. pr. 8. B Iambi, op. cit., p. 10. 17.
6 Nicom. i. 7. 1. 7 Metaph. A. 13, 1020 a 13.
8 lb. I. 1, 1053 a 30 ; Z. 13. 1039 a 12.
H lb. M. 9, 1085 b 22. 10 Phijs. iii. 7, 207 b 7.
11 Metaph. I. 6, 1057 a 3. 12 lb. N. 1. 1088 a 5.
13 Stob. Eel. i. 21. 7 C (Vo>s. i 3 , p. 310. 11-14). 14 Nicom. i. 7. 3.
CLASSIFICATION OF NUMBERS 71
both odd or both even); while an odd number is that which,
however divided, must in any case fall into two unequal parts,
and those parts always belonging to the two different kinds
respectively (i.e. one being odd and one even).' 1
In the latter definition we have a trace of the original
conception of 2 (the dyad) as being, not a number at all, but
the principle or beginning of the even, just as one was not a
number but the principle or beginning of number ; the defini-
tion implies that 2 was not originally regarded as an even
number, the qualification made by Nicomachus with reference
to the dyad being evidently a later addition to the original
definition (Plato already speaks of two as even). 2
With regard to the term ' odd-even ', it is to be noted that ?
according to Aristotle, the Pythagoreans held that ' the One
arises from both kinds (the odd and the even), for it is both
even and odd \ 3 The explanation of this strange view might
apparently be that the unit, being the principle of all number,
even as well as odd, cannot itself be odd and must therefore
be called even-odd. There is, however, another explanation,
attributed by Theon of Smyrna to Aristotle, to the effect that the
unit when added to an even number makes an odd number, but
when added to an odd number makes an even number : which
could not be the case if it did not partake of both species ;
Theon also mentions Archytas as being in agreement with this
view. 4 But, inasmuch as the fragment of Philolaus speaks of
' many forms ' of the species odd and even, and ' a third '
(even-odd) obtained from a combination of them, it seems
more natural to take ' even-odd ' as there meaning, not the
unit, but the product of an odd and an even number, while, if
' even ' in the same passage excludes such a number, ' even '
would appear to be confined to powers of 2, or 2 n .
We do not know how far the Pythagoreans advanced
towards the later elaborate classification of the varieties of
odd and even numbers. But they presumably had not got
beyond the point of view of Plato and Euclid. In Plato we
have the terms f even-times even ' (dpria dpriaKis), ' odd-
times odd' (nepLTTa 7r€piTTaKis), i odd-times even' (dpria
1 Nicom. i. 7. 4. 2 Plato, Parmenides, 143 D.
3 Arist. Metaph. A. 5, 986 a 19.
4 Theon of Smyrna, p. 22. 5-10.
72 PYTHAGOREAN ARITHMETIC
7T€pLTTdKis) and ' even-times odd' (nepiTra dpriaKis), which
are evidently used in the simple sense of the products of even
and even, odd and odd, odd and even, and even and odd
factors respectively. 1 Euclid's classification does not go much
beyond this ; he does not attempt to make the four defini-
tions mutually exclusive. 2 An ' odd-times odd ' number is of
course any odd number which is not prime ; but ' even-times
even ' (' a number measured "by an even number according to
an even number ') does not exclude ' even-times odd ' (' a
number measured by an even number according to an odd
number'); e.g. 24, which is 6 times 4, or 4 times 6, is also
8 times 3. Euclid did not apparently distinguish, any more
than Plato, between * even-times odd ' and ' odd-times even '
(the definition of the latter in the texts of Euclid was pro-
bably interpolated). The Neo-Pythagoreans improved the
classification thus. With them the ' even- times even ' number
is that which has its halves even, the halves of the halves
even, and so on till unity is reached ' 3 ; in short, it is a number
of the form 2 n . The ' even-odd ' number (apTioirepLTTOs in one
word) is such a number as, when once halved, leaves as quo-
tient an odd number, 4 i.e. a number of the form 2(2m+l).
The ' odd-even ' number {TTtpio-crdpTios) is a number such that
it can be halved twice or more times successively, but the
quotient left when it can no longer be halved is an odd num-
ber not unity, 5 i.e. it is a number of the form 2 n+1 (2m + 1).
The ' odd-times odd ' number is not defined as such by
Nicomachus and Iamblichus, but Theon of Smyrna quotes
a curious use of the term ; he says that it was one of the
names applied to prime numbers (excluding of course 2), for
these have two odd factors, namely 1 and the number itself.
Prime or incomposite numbers (wpcoros /ecu avvv Octos) and
secondary or composite numbers (Sevrepos kcu avvOtros) are
distinguished in a fragment of Speusippus based upon works
of Philolaus. 7 We are told 8 that Thymaridas called a prime
number rectilinear (euOvypa/ifjLiKos), the ground being that it
can only be set out in one dimension 9 (since the only measure
1 Plato, Parmenides, 143 E. ' 2 See Eucl. VII. Defs. 8-10.
3 Nicom. i. 8. 4. 4 lb. i. 9. 1. 5 lb. i. 10. 1.
,; Theon of Smyrna, p. 23. 14-23.
7 Theol. Ar. (Ast), p. 62 (Vors. i 8 , p. 304. 5).
* Iambi, in Nicom., p. 27. 4. 9 Cf. Arist. Metaph. A. 13, 1020 b 3, 4.
CLASSIFICATION OF NUMBERS 73
of it, excluding the number itself, is 1); Theon of Smyrna
gives euthymetric and linear as alternative terms, 1 and the
latter (ypafifiLKos) also occurs in the fragment of Speusippus.
Strictly speaking, the prime number should have been called
that which is rectilinear or linear only. As we have seen,
2 was not originally regarded as a prime number, or even as
a number at all. But Aristotle speaks of the dyad as ' the
only even number which is prime,' 2 showing that this diver-
gence from early Pythagorean doctrine took place before
Euclid's time. Euclid defined a prime number as ' that which
is measured by a unit alone ', 3 a composite number as ' that
which is measured by some number ', 4 while he adds defini-
tions of numbers ' prime to one another ' (' those which are
measured by a unit alone as a common measure ') and of
numbers ' composite to one another ' (' those which are mea-
sured b}^ some number as a common measure '). 5 Euclid then,
as well as Aristotle, includes 2 among prime numbers. Theon
of "Smyrna says that even numbers are not measured by the
unit alone, except 2, which therefore is odd-^&e without being
prime. 6 The Neo-Pythagoreans, Nicomachus and Iamblichus,
not only exclude 2 from prime numbers, but define composite
numbers, numbers prime to one another, and numbers com-
posite to one another as excluding all even numbers ; they
make all these categories subdivisions of odd? Their object
is to divide odd into three classes parallel to the three subdivi-
sions of even, namely even-even = 2 W , even-odd = 2 (2m + 1)
and the quasi-intermediate odd-even = 2 n+1 (2m+ 1) ; accord-
ingly they divide odd numbers into (a) the prime and
incomposite, which are Euclid's primes excluding 2, (6) the
secondary and composite, the factors of which must all be not
only odd but prime numbers, (c) those which are ' secondary and
composite in themselves but prime and incomposite to another
number,' e.g. 9 and 25, which are both secondary and com-
posite but have no common measure except 1. The incon-
venience of the restriction in (b) is obvious, and there is the
1 Theon of Smyrna, p. 23. 12.
2 Arist. Topics, e. 2, 157 a 39.
3 Eucl. VII. Def. 11. 4 lb. Def, 13.
B lb. Defs. 12, 14.
6 Theon of Smyrna, p. 24. 7.
7 Nicom. i, cc. 11-13 ; Iambi, in Nicom., pp. 26-8.
74 PYTHAGOREAN ARITHMETIC
further objection that (b) and (c) overlap, in fact (6) includes
the whole of (c).
' Perfect ' and ' Friendly ' numbers.
There is no trace in the fragments of Philolaus, in Plato or
Aristotle, or anywhere before Euclid, of the perfect number
(reXecos) in the well-known sense of Euclid's definition
(VII. Def. 22), a number, namely, which is ' equal to (the
sum of) its own parts' (i.e. all its factors including 1),
e.g. 6=1+2 + 3; 28 = 1+2 + 4 + 7 + 14;
496 = 1 +2 + 4 + 8 + 16 + 31+62 + 124 + 248.
The law of the formation of these numbers is proved in
Eucl. IX. 36, which is to the effect that, if the sum of any
number of terms of the series 1 , 2, 2 2 , 2 3 . . . . 2 n_1 ( = S n ) is prime,
then S n . 2 n ~ l is a ' perfect ' number. Theon of Smyrna * and
Nicomachus 2 both define a ' perfect ' number and explain the
law of its formation ; they further distinguish from it two
other kinds of numbers, (1) over-perfect (v-rrepTeXrjs or vntpTe-
Aefoy), so called because the sum of all its aliquot parts is
greater than the number itself, e.g. 12, which is less than
1 + 2 + 3 + 4 + 6, (2) defective (eWiTrrjs), so called because the
sum of all its aliquot parts is less than the number itself,
e. g. 8, which is greater than 1+2 + 4. Of perfect numbers
Nicomachus knew four (namely 6. 28, 4^6, 8128) but no more.
He says they are formed in ' ordered ' fashion, there being one
among the units (i.e. less than 10), one among the tens (less
than 100), one among the hundreds (less than 1000), and one
among the thousands (less than a myriad) ; he adds that they
terminate alternately in 6 or 8. They do all terminate in 6 or
8 (as we can easily prove by means of the formula (2 n — 1) 2 n ~ l ) i
but not alternately, for the fifth and sixth perfect numbers
both end in 6, and the seventh and eighth both end in 8.
Iamblichus adds a tentative suggestion that there may (el
Tvyoi) in like manner be one perfect number among the first
myriads (less than 10000 2 ), one among the second myriads
(less than 10000 3 ), and so on ad infinitum? This is incorrect,
for the next perfect numbers are as follows : 4
1 Theon of Smyrna, p. 45. 2 Nicom. i. 16, 1-4.
:? Tambl. in Nicom., p. 33. 20-23.
4 The fifth perfect number may have been known to Iamblichus,
'PERFECT' AND 'FRIENDLY' NUMBERS 75
fifth, 2 12 (2 13 -1) = 33 550 336
sixth, 2 1C (2 17 -1) = 8 589 809 056
seventh, 2 18 (2 19 -1) = 137 438 691 328
eighth, 2 30 (2 31 -1) == 2 305 843 008 139 952 128
ninth, 2 60 (2 G1 -1) = 2 658 455 991 569 831 744 654 692
615 953 842 176
tenth, 2 s8 (2 89 -l).
With these ' perfect ' numbers should be compared the so-
called ' friendly numbers '. Two numbers are ' friendly ' when
each is the sum of all the aliquot parts of the other, e.g. 284 and
220 (for 284 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110,
while 220 = 1+2 + 4 + 71 + 142). Iamblichus attributes the
discovery of such numbers to Pythagoras himself, who, being
asked ' what is a friend ? ' said 'Alter ego ', and on this analogy
applied the term ' friendly ' to two numbers the aliquot parts
of either of which make up the other. 1
While for Euclid, Theon of Smyrna, and the Neo-Pytha-
goreans the ' perfect ' number was the kind of number above
described, we are told that the Pythagoreans made 10 the
perfect number. Aristotle says that this was because they
found within it such things as the void, proportion, oddness,
and so on. 2 The reason is explained more in detail by Theon
of Smyrna 3 and in the fragment of Speusippus. 10 is the
sum of the numbers 1, 2, 3, 4 forming the rerpaKrvs (' their
greatest oath ', alternatively called the ' principle of health ' 4 ).
These numbers include the ratios corresponding to the musical
intervals discovered by Pythagoras, namely 4 : 3 (the fourth),
though he does not give it ; it was, however, known, with all its factors,
in the fifteenth century, as appears from a tract written in German
which was discovered by Curtze (Cod. lat. Monac. 14908). The first
eight 'perfect' numbers were calculated by Jean Prestet (d. 1670);
Fermat (1601-65) had stated, and Kuler proved, that 2 3l -l is prime.
The ninth perfect number was found by P. Seelhoff, Zeitschr.f. Math. u.
Physik, 1886, pp. 174 sq.) and verified by E. Lucas (Mathesis, vii, 1887,
pp. 44-6). The tenth was found by R. E. Powers (Bull. Amer. Math.
Soc, 1912, p. 162).
1 Iambi, in Nicom., p. 35. 1-7. The subject of 'friendly' numbers
was taken up by Euler, who discovered no less than sixty-one pairs of
such numbers. Descartes and van Schooten had previously found three
pairs but no more.
2 Arist. Metaph. M. 8, 1084 a 32-4.
3 Theon of Smyrna, p. 93. 17-94. 9 [Vorsokratiker, i 3 , pp. 303-4).
4 Lucian, De lapsu in salutando, 5.
76 PYTHAGOREAN ARITHMETIC
3 : 2 (the fifth), and 2 : 1 (the octave). Speusippus observes
further that 10 contains in it the 'linear', 'plane' and 'solid'
varieties of number ; for 1 is a point, 2 is a line, 1 3 a triangle,
and 4 a pyramid. 2
Figured numbers.
This brings us once more to the theory of figured numbers,
which seems to go back to Pythagoras himself. A point or
dot is used to represent 1 ; two dots placed apart represent
2, and at the same time define the straight line joining the
two dots ; three dots, representing 3, mark out the first
rectilinear plane figure, a triangle ; four dots, one of which is
outside the plane containing the other three, represent 4 and
also define the first rectilineal solid figure. It seems clear
that the oldest Pythagoreans were acquainted with the forma-
tion of triangular and square numbers by means of pebbles or
dots 3 ; and we judge from the account of Speusippus's book,
On the Pythagorean Numbers, which was based on works of
Philolaus, that the latter dealt with linear numbers, polygonal
numbers, and plane and solid numbers of all sorts, as well as
with the five regular solid figures. 4 The varieties of plane
numbers (triangular, square, oblong, pentagonal, hexagonal,
and so on), solid numbers (cube, pyramidal, &c.) are all dis-
cussed, with the methods of their formation, by Nicomachus 5
and Theon of Smyrna. 6
(a) Triangular numbers.
To begin with triangular numbers. It was probably
Pythagoras who discovered that the sum of any number of
successive terms of the series of natural numbers 1, 2, 3 . . .
beginning from 1 makes a triangular number. This is obvious
enough from the following arrangements of rows of points ;
Thus 1 + 2 + 3 + . . . + n = ^n (n + I) is a triangular number
1 Cf. Arist. Metaph. Z. 10, 1036 b 12. 2 Theol. Ar. (Ast), p. 62. 17-22.
3 Cf. Arist. Metaph. N. 5, 1092 b 12. 4 Theol. Ar. (Ast), p. 61.
5 Nicom. i. 7-11, 13-16, 17. 6 Theon of Smyrna, pp. 26-42.
FIGURED NUMBERS 77
of side n. The particular triangle which has 4 for its side is
mentioned in a story of Pythagoras by Lucian. Pythagoras
told some one to count. He said 1, 2, 3, 4, whereon Pytha-
goras interrupted, 'Do you see ? What you take for 4 is 10,
a perfect triangle and our oath'. 1 This connects the know-
ledge of triangular numbers with true Pythagorean ideas.
(/?) Square numbers and gnomons.
We come now to square numbers. It is easy to see that, if
we have a number of dots forming and filling
up a square as in the accompanying figure reprer
senting 16, the square of 4, the next higher
square, the square of 5, can be formed by adding
a row of dots round two sides of the original
square, as shown ; the number of these dots is
2 . 4 + 1, or 9. This process of forming successive squares can
be applied throughout, beginning from the first square
number 1. The successive additions are shown in the annexed
figure between the successive pairs of straight
lines forming right angles ; and the succes-
sive numbers added to the 1 are
_J. .
1
.
3, 5. 7 ... (271+1),
that is to say, the successive odd numbers.
This method of formation shows that the
sum of any number of successive terms
of the series of odd numbers 1, 3, 5, 7 . . . starting from
1 is a square number, that, if n 2 is any square number, the
addition of the odd number 2^+1 makes it into the next
square, (?i+l) 2 , and that the sum of the series of odd num-
bers 1+3 + 5 + 7 + .. .+ (2n+l) = (n + l) 2 , while
1 + 3 + 5 + 7 + .. . + (2^-1) = n 2 .
All this was known to Pythagoras. The odd numbers succes-
sively added were called gnomons ; this is clear from Aristotle's
allusion to gnomons placed round 1 which now produce different
figures every time (oblong figures, each dissimilar to the pre-
ceding one), now preserve one and the same figure (squares) 2 ;
the latter is the case with the gnom'ons now in question.
1 Lucian, Bia>v npaats, 4. 2 Arist. Phys. iii. 4, 203 a 13-15.
78 PYTHAGOREAN ARITHMETIC
(y) History of the term ' gnomon \
It will be noticed that the gnomons shown in the above
figure correspond in shape to the geometrical gnomons with
which Euclid, Book II, has made us familiar. The history of
the word ' gnomon ' is interesting. (1) It was originally an
astronomical instrument for the measuring of time, and con-
sisted of an upright stick which cast shadows on a plane or
hemispherical surface. This instrument is said to have been
introduced into Greece by Anaximander 1 and to have come
from Babylon. 2 Following on this application of the word
' gnomon ' (a ' marker ' or ' pointer ', a means of reading off and
knowing something), we find Oenopides calling a perpendicular
let fall on a straight line from an external point a straight line
drawn ' gnomon-wise ' (Kara yycofioua). 3 Next (2) we find the
term used of an instrument for drawing right angles, which
took the form shown in the annexed figure. This seems to
be the meaning in Theognis 805, where it is said
that the envoy sent to consult the oracle at Delphi
should be ' straighter than the ropvos (an instru-
ment with a stretched string for drawing a circle),
the o-rdOfirj (a plumb-line), and the gnomon'.
It was natural that, owing to its shape, the gnomon should
then be used to describe (3) the figure which remained of
a square when a smaller square was cut out of it (or the figure
which, as Aristotle says, when added to a square, preserves
the shape and makes up a larger square). The term is used
in a fragment of Philolaus where he says that ' number makes
all things knowable and mutually agreeing in the way charac-
teristic of the gnomon '. 4 Presumably, as Boeckh says, the
connexion between the gnomon and the square to which it is
added was regarded as symbolical of union and agreement,
and Philolaus used the idea to explain the knowledge of
things, making the knowing embrace the known as the
gnomon does the square. 5 (4) In Euclid the geometrical
meaning of the word is further extended (II. Def. 2) to cover
1 Suidas, s.v. 2 Herodotus, ii. 109.
3 Proclus on Eucl. I, p. 283. 9.
4 Boeckh, Philolaos des Pythagoreers Lehren, p. 141 ; ib., p. 144 ; Vors. i 3 ,
p. 313. 15.
5 Cf. Scholium No. 11 to Book II in Euclid, ed. Heib., vol. v, p. 225.
HISTORY OF THE TERM 'GNOMON
j >
79
the figure similarly related to any parallelogram ; instead of
a square ; it is defined as made up of ' any
one whatever of the parallelograms about
the diameter (diagonal) with the two com-
plements '. Later still (5) Heron of Alex-
andria defines a gnomon in general as that
which, when added to anything, number or figure, makes the
whole similar to that to which it is added. 1
(8) Gnomons of the polygonal numbers.
Theon of Smyrna uses the term in this general sense with
reference to numbers : ' All the successive numbers which [by
being successively added] produce triangles or squares or
polygons are called gnomons.' 2 From the accompanying
figures showing successive pentagonal and hexagonal numbers
it will be seen that the outside rows or gnomons to be succes-
sively added after 1 (which is the first pentagon, hexagon, &c.)
are in the case of the pentagon 4, 7, 10 , . . or the terms of an
arithmetical progression beginning from 1 with common differ-
ence 3, and in the case of the hexagon 5, 9, 13 .... or the
terms of an arithmetical progression beginning from 1 with
common difference 4. In general the successive gnomonic
numbers for any polygonal number, say of n sides, have
(n — 2) for their common difference. 3
(e) Right-angled triangles with sides in rational numbers.
To return to Pythagoras. Whether he learnt the fact from
Egypt or not, Pythagoras was certainly aware that, while
3 2 + 4 2 = 5 2 , any triangle with its sides in the ratio of the
1 Heron, Def. 58 (Heron, vol. iv, Heib., p. 225).
2 Theon of Smyrna, p. 37. 11-13. 3 lb., p. 34. 13-15.
80 PYTHAGOREAN ARITHMETIC
numbers 3, 4, 5 is right angled. This fact could not but add
strength to his conviction that all things were numbers, for it
established a connexion between numbers and the angles of
geometrical figures. It would also inevitably lead to an
attempt to find other square numbers besides 5 2 which are
the sum of two squares, or, in other words, to find other sets
of three integral numbers which can be made the sides of
right-angled triangles ; and herein we have the beginning of
the indeterminate analysis which reached so high a stage of
development in Diophantus. In view of the fact that the
sum of any number of successive terms of the series of odd
numbers 1, 3, 5, 7 . . . beginning from 1 is a square, it was
only necessary to pick out of this series the odd numbers
which are themselves squares; for if we take one of these,
say 9, the addition of this square to the square which is the sum
of all the preceding odd numbers makes the square number
which is the sum of the odd numbers up to the number (9) that
we have taken. But it would be natural to seek a formula
which should enable all the three numbers of a set to be imme-
diately written down, and such a formula is actually attributed
to Pythagoras. 1 This formula amounts to the statement that,
if m be any odd number,
_ /m 2 — K 2 /wi 2 +K 2
m +(-^—) =(— f~) •
Pythagoras would presumably arrive at this method of forma-
tion in the following way. Observing that the gnomon put
round n 2 is 2n+l, he would only have to make 2 n + 1 a
square.
If we suppose that 2 n + 1 = m 2 ,
we obtain n = \ (m 2 — 1),
and therefore n + 1 = -| (m 2 +1).
It follows that
m-
,m 2 — 1 \ 2 ,m 2 + 1 \ 2
V 9 / z \ o )
1 Proclus on Eucl. I, p. 487. 7-21.
RATIONAL RIGHT-ANGLED TRIANGLES 81
Another formula, devised for the same purpose, is attributed
to Plato, 1 namely
(2m) 2 + (m 2 - l) 2 == (m 2 + l) 2 .
We could obtain this formula from that of Pythagoras by
doubling the sides of each square in the latter ; but it would
be incomplete if so obtained, for in Pythagoras' s formula m is
necessarily odd, whereas in Plato's it need not be. As Pytha-
goras's formula was most probably obtained from the gnomons
of dots, it is tempting to suppose that Plato's was similarly
evolved. Consider the square with n dots in its
side in relation to the next smaller square (n — 1 ) 2 - — f*
and the next larger (n + l) 2 . Then n 2 exceeds . .
(n— l) 2 by the gnomon 2n—l, but falls short of » ■
(n+l) 2 by the gnomon 2 n + l. Therefore the -J 1_
square (n+1) 2 exceeds the square (n— l) 2 by
the sum of the two gnomons 2n — 1 and 2n + l, which
is 4n.
That is, 4n + (n-l) 2 = (n+l) 2 ,
and, substituting m 2 for n in order to make 4iu square, we
obtain the Platonic formula
(2m) 2 + (m 2 — 1) 2 = (m 2 +l) 2 .
The formulae of Pythagoras and Plato supplement each
other. Euclid's solution (X, Lemma following Prop. 28) is
more general, amounting to the following.
If AB be a straight line bisected at C and produced to D,
then (Eucl. II. 6)
AD.DB + CB 2 = CD 2 ,
which we may write thus :
uv = c 2 — b 2 ,
where u — c + b, v = c — b,
and consequently
c = %(u + v) } 6=-|(u — v).
In order that uv may be a square, says Euclid, u and v
must, if they are not actually squares, be ' similar plane num-
bers ', and further they must be either both odd or both even
1 Proclus on Eucl. I, pp.428. 21-429. 8.
1523 Q
82 PYTHAGOREAN ARITHMETIC
in order that b (and c also) may be a whole number. ' Similar
plane ' numbers are of course numbers which are the product
of two factors proportional in pairs, as Trip, np and mq. nq, or
mnp 2 and mivf. Provided, then, that these numbers are both
even or both odd,
o o o o ,mnp 2 — mnq\ 2 ,mnp 2 + mnq 2 \ 2
m A n l p z q 2 + i — — —J = ( — — J
is the solution, which includes both the Pythagorean and the
Platonic formulae.
(£) Oblong numbers.
Pythagoras, or the earliest Pythagoreans, having discovered
that, by adding any number of successive terms (beginning
from 1) of the series 1 + 2 + 3 + ... + n — \n (n + 1), we obtain
triangular numbers, and that by adding the successive odd
numbers 1 + 3 + 5 + ... +(2n— 1) = n 2 we obtain squares, it
cannot be doubted that in like manner they summed the
series of even numbers 2 + 4 + 6 + . . . + 2 n = n (n + 1) and
discovered accordingly that the sum of any number of succes-
sive terms of the series beginning with 2 was an ' oblong '
number (iTepofirJKrj^), with ' sides ' or factors differing by 1.
They would also see that the oblong number is double of
a triangular number. These facts would be brought out by
taking two dots representing 2 and then placing round them,
gnomon-wise and successively, the even numbers 4^ 6, &c,
thus :
t
The successive oblong numbers are
2.3 = 6, 3.4 = 12, 4.5 = 20..., n(n+l) ...,
and it is clear that no two of these numbers are similar, for
the ratio n:(n+l) is different for all different values of n.
We may have here an explanation of the Pythagorean identi-
fication of ' odd ' with ' limit ' or ' limited ' and of ' even ' with
OBLONG NUMBERS 83
' unlimited ' l (cf. the Pythagorean scheme of ten pairs of
opposites, where odd, limit and square in one set are opposed
to even, unlimited and oblong respectively in the other). 2 For,
while the adding of the successive odd numbers as gnomons
round 1 gives only one form, the square, the addition of the
successive even numbers to 2 gives a succession of ' oblong '
numbers all dissimilar in form, that is to say, an infinity of
forms. This seems to be indicated in the passage of Aristotle's
Physics where, as an illustration of the view that the even
is unlimited, he says that, where gnomons are put round 1,
the resulting figures are in one case always different in
species, while in the other they always preserve one form 3 ;
the one form is of course the square formed by adding the
odd numbers as gnomons round 1 ; the words kolI yoopfc
(' and in the separate case ', as we may perhaps translate)
imperfectly describe the second case, since in that case
even numbers are put round 2, not 1, but the meaning
seems clear. 4 It is to be noted that the word erepo/jLiJKrj?
(' oblong ') is in Theon of Smyrna and Nicomachus limited to
numbers which are the product of two factors differing by
unity, while they apply the term 7rpo/j,rJKr]9 ('prolate', as it
were) to numbers which are the product of factors differing
by two or more (Theon makes TrpofirJKrjs include irepofjLrJKrj?).
In Plato and Aristotle irepo/irJKrjs has the wider sense of any
non-square number with two unequal factors.
It is obvious that any 'oblong' number n(n-\-l) is the
sum of two equal triangular numbers. Scarcely less obvious
is the theorem of Theon that any square number is made up
of two triangular numbers 5 ; in this case, as is seen from the
1 Arist. Metaph. A. 5, 986 a 17.
2 lb. A. 5, 986 a 23-26.
3 Arist. Phys. iii. 4, 203 a 10-15.
4 Cf. Plut. (?) Stob. Ed. i. pr. 10, p. 22. 16 Wachsmuth.
5 Theon of Smyrna, *p. 41. 3-8.
a 2
84 PYTHAGOREAN ARITHMETIC
figure, the sides of the triangles differ by unity, and of course
/ \n{n— l) + \n(n + 1) = n 2 .
• • -/> • Another theorem connecting triangular num-
• y % * ' bers and squares, namely that 8 times any
Y* triangular number + 1 makes a square, may
easily go "back to the early Pythagoreans. It is
quoted by Plutarch 1 and used by Diophantus, 2 and is equi-
valent to the formula
&.%n(n+l) + l = 4n(n+ 1) + 1 = (2n+l) 2 .
It may easily have been proved by means of a figure
made up of dots in the usual way. Two
. , equal triangles make up an oblong figure
• • of the form n(n+l), as above. Therefore
we have to prove that four equal figures
. . of this form with one more dot make up
. . (2^+l) 2 . The annexed figure representing
• • 7 2 shows how it can be divided into four
' oblong ' figures 3 . 4 leaving 1 over.
In addition to Speusippus, Philippus of Opus (fourth
century), the editor of Plato's Laws and author of the Epi-
nomis, is said to have written a work on polygonal numbers. ;i
Hypsicles, who wrote about 170 B.C., is twice mentioned in
Diophantus's Polygonal Numbers as the author of a ' defini-
tion ' of a polygonal number.
The theory of proportion and means.
The ' summary ' of Proclus (as to which see the beginning
of Chapter IV) states (if Friedlein's reading is right) that
Pythagoras discovered ' the theory of irrationals (rrju tcdv
dXoycov irpayiiartiav) and the construction of the cosmic
figures' (the five regular solids). 4 We are here concerned
with the first part of this statement in so far as the reading
aXoyoav (' irrationals ') is disputed. Fabricius seems to have
been the first to record the variant dvaXoycov, which is also
noted by E. F. August 5 ; Mullach adopted this reading from,
1 Plutarch, Plat. Quaest. v. 2. 4, 1003 F. 2 Dioph. IV. 38.
3 Btoypdcfroi, Vitarum scriptores Graeci mlnores, ed. Westermann, p. 446.
4 Proclus on Eucl. I, p. 65. 19.
5 In his edition of the Greek text of Euclid (1824-9), vol. i, p. 290.
THE THEORY OF PROPORTION AND MEANS 85
Fabricius. avaXoycov is not the correct form of the word, but
the meaning would be ' proportions ' or ' proportionals ', and
the true reading may be either touv avoCkoyicov (' proportions '),
or, more probably, rcov ava Xoyov (' proportionals ') ; Diels
reads tcov ava Xoyov, and it would seem that there is now
general agreement that dXoyoov is wrong, and that the theory
which Proclus meant to attribute to Pythagoras is the theory ,
ot proportion or proportionals, not of irrationals.
(a) Arithmetic, geometric, and harmonic means.
It is true that we have no positive evidence of the use by
Pythagoras of proportions in geometry, although he must
have been conversant with similar figures, which imply some
theory of proportion. But he discovered the dependence of
musical intervals on numerical ratios, and the theory of means
was developed very early in his school with reference to
the theory of music and arithmetic. We are told that in
Pythagoras's time there were three means, the arithmetic,
the geometric, and the subcontrary, and that the name of the
third (' subcontrary ') was changed by Archytas and Hippasus
to 'harmonic \ l A fragment of Archytas's work On Music
actually defines the three ; we have the arithmetic mean
when, of three terms, the first exceeds the second by the
same amount as the second exceeds the third ; the geometric
mean when, of the three terms, the first is to the second as
the second is to the third ; the ' subcontrary, which we call
harmonic ', when the three terms are such that ' by whatever
part of itself the first exceeds the second, the second exceeds
the third by the same part of the third \ 2 That is, if a, b, c
a
are in harmonic progression, and a = b 4- - 3 we must have
b = c + - -j whence in fact
n
a a—b 1 1 1 1
= t— > or - — 7
c b—c c b b a
Nicomachus too says that the name ' harmonic mean ' was
adopted in accordance with the view of Philolaus about the
' geometrical harmony ', a name applied to the cube because
it has 12 edges, 8 angles, and 6 faces, and 8 is the mean
1 Iambi, in Nicom., p. 100. 19-24.
2 Porph. in Ptol. Harm., p. 267 {Vors. i 3 , p. 334. 17 sq.).
86 PYTHAGOREAN ARITHMETIC
between 12 and 6 according to the theory of harmonics (Kara
tt]v apixoviKr)v)}
Iamblichus, 2 after Nicomachus, 3 mentions a special ' most
perfect proportion ' consisting of four terms and called
' musical ', which, according to tradition, was discovered by
the Babylonians and was first introduced into Greece by
Pythagoras. It was used, he says, by many Pythagoreans,
e. g. (among others) Aristaeus of Croton, Timaeus of Locri,
Philolaus and Archytas of Tarentum, and finally by Plato
in the Timaeus, where we are told that the double and triple
intervals were filled up by two means, one of which exceeds
and is exceeded by the same .part of the extremes (the
harmonic mean), and the other exceeds and is exceeded by
the same numerical magnitude (the arithmetic mean). 4 The
proportion is
a + b lab 7
a : = = : b,
2 a + b
an example being 12:9 = 8:6.
(/5) Seven other means distinguished.
The theory of means was further developed in the school
by the gradual addition of seven others to the first three,
making ten in all. The accounts of the discovery of the
fourth, fifth, and sixth are not quite consistent. In one place
Iamblichus says they were added by Eudoxus 5 ; in other
places he says they were in use by the successors of Plato
down to Eratosthenes, but that Archytas and Hippasus made
a beginning with their discovery, or that they were part of
the Archytas and Hippasus tradition. 7 The remaining four
means (the seventh to the tenth) are said to have been added
by two later Pythagoreans, Myonides and Euphranor. 8 From
a remark of Porphyry "it would appear that one of the first
seven means was discovered by Simus of Posidonia, but
that the jealousy of other Pythagoreans would have robbed
him of the credit. 9 The ten means are described by
1 Nicom. ii. 26. 2. 2 Iambi, in Nicom., p. 118. 19 sq.
3 Nicom. ii. 29. 4 Plato, Timaeus, 36 A.
5 Iambi, in Nicom., p. 101. 1-5. 6 lb., p. 116. 1-4.
7 lb., p. 113, 16-18. 8 lb., p. 116. 4-6.
9 Porphyry, Vit. Pyth. 3 ; Vors. i 3 , p. 343. 12-15 and note.
THE SEVERAL MEANS DISTINGUISHED 87
Nicomachus * and Pappus 2 ; their accounts only differ as
regards one of the ten. If a>b>c, the formulae in the third
column of the following table show the various means.
Equivalent.
a + c — 2 b (arithmetic)
ac = b 2 (geometric)
1 1 2 /i.
— h- = T (harmonic)
a c b v '
a 2 + c 2 , (subcontrary to
a + c harmonic)
1 u. in
XX U. Ill
Pappus.
Formulae.
icom.
a— b a b c
1
1
b — c a b ~~ c
2
2
a — b a\' bl
b-c ~ bl~c\
a — b a
3
3
b — c c
a — b c
4
4
b — c a
a — b c
5
5
b^c = b
•
a — b b
6
6
b — c "~ a
a — c a
7
(omitted)
b — c " c
a — c a
8
9
a — b c
a — c b
9
10
b—c c
a — c b
10
7
l '
•
a — b c
a—c a
nitted
) 8
a — b b
a
7 C \
h + C ~b
c = a + b —
a-
(subcontrary
to geometric)
b)
c 2 = 2ac — ab
a 2 + c 2 = a(b + c)
b 2 + c 2 = c(a + b)
a — b + c
a 2 = lab— be
The two lists together give jive means in addition to the
first six which are common to both : there would be six more
a — c
a
(as Theon of Smyrna says 3 ) were it not that y = - is
illusory, since it gives a = b. Tannery has remarked that
1 Nicom. ii. 28.
3 Theon of Smyrna, p. 106. 15, p. 116. 3.
Pappus, iii, p. 102.
88 PYTHAGOREAN ARITHMETIC
Nos. 4, 5, 6 of the above means give equations of the second
degree, and he concludes that the geometrical and even the
arithmetical solution of such equations was known to the dis-
coverer of these means, say about the time of Plato l ; Hippo-
crates of Chios, in fact, assumed the geometrical solution of
a mixed quadratic equation in his quadrature of lunes.
Pappus has an interesting series of propositions with
regard to eight out of the ten means defined by him. 2 He
observes that if oc, p, y be three terms in geometrical pro-
gression, we can form from these terms three other terms
a, b, c, being linear functions of a, P, y which satisfy respec-
tively eight of the above ten relations ; that is to say, he
gives a solution of eight problems in indeterminate analysis
of the second degree. The solutions are as follows :
No. in
Nicom.
No. in
Pappus.
Formulae.
Solution
in terms of
a, fi, y.
Smallest
solution.
2
2
a — 6 a b
b — c b c
a= a + 2/? + y
b= p + y
y
a = 4
b = 2
c = 1
3
3
a — b a
b — c "~ c
a = 2a + 3/3 + y
6 = 2(3 + y
c = /3 + y
a— 6
6 = 3
c = 2
4
4
a — b c
b — c a
a = 2a+30+y
b — 2a + 2/3 + y
c = fi + y
*
a = 6
b = 5
c =2
5
5
a—b c
b-c = b
a= a + 3/? + y
b — a+2p+y
c = p + y
a — 5
6 = 4
c = 2
6
G
a—b b
b— c a
a = a + 3/3 + 2y
b = a + 2/5+ y
c = a+ P— y
a = 6
6 = 4
c = 1
1 Tannery, Memoires scienfijiqites, i, pp. 92-3.
2 Pappus, iii, pp. 84-104.
THE SEVERAL MEANS DISTINGUISHED 89
No in No. in w v • ± c Smallest
±>u. _l u Formulae. in terms of a nl«+i«n
Nicom. Pappus. o solution.
ex, p, y.
a — c__a a = 2oc + 3p + y a = G
a-b ~~~b b = a + 2/3 + y b = 4
c = 2/S + y c = 3
8 9 a ~ c - a ' a= x + zP + y a — 4
a-b "c b= (X+ /3 + y b = 3
c = fi + y c = 2
a — c ft a=a+/3 + y a = 3
9 10 /~~c 6= £ + y
c = y
c = 1
Pappus does not include a corresponding solution for his
No. 1 and No. 7, and Tannery suggests as the reason for this
that, the equations in these cases being already linear, there
is no necessity to assume ay = /3 2 , and consequently there is
one indeterminate too many. 1 Pappus does not so much prove
,.' a
as verify his results, by transforming the proportion -= = -
in all sorts of ways, componendo, dividendo, &c.
(y) Plato on geometric means between two squares
or tivo cubes.
It is well known that the mathematics in Plato's Timaeus
is essentially Pythagorean. It is therefore a priori probable
that Plato TTvOayopifct, in the passage 2 where he says that
between two planes one mean suffices, but to connect two
solids two means are necessary. By planes and solids he
really means square and cube numbers, and his remark is
equivalent to stating that, if p 2 , q 2 are two square numbers,
p 2 : pq = pq : q 2 ,
while, if p 3 , q 3 are two cube numbers,
p 3 : p 2 q = p 2 q : pq 2 = pq 2 : q 3 ,
the means being of course means in continued geometric pro-
portion. Euclid proves the properties for square and cube
1 Tannery, loc. cit., pp. 97-8. 2 Plato, Timaeus, 32 A, E.
90 PYTHAGOREAN ARITHMETIC
numbers in VIII. 11, 12, and for similar plane and solid num-
bers in VIII. 18, 19. Nicomachus quotes the substance of
Plato's remark as a ' Platonic theorem ', adding in explanation
the equivalent of Eucl. VIII. 11, 12. 1
(8) A theorem of Archytas.
Another interesting theorem relative to geometric means
evidently goes back' to the Pythagoreans. If we have two
numbers in the ratio known as twiuopLos, or super particular is,
i.e. the ratio of n+\ to n, there can be no number which is
a mean proportional between them. The theorem is Prop. 3 of
Euclid's Sectio Canonist and Boetius has preserved a proof
of it by Archytas, which is substantially identical with that of
Euclid. 3 The proof will be given later (pp. 215-16). So far as
this chapter is concerned, the importance of the proposition lies
in the fact that it implies the existence, at least as early
as the date of Archytas (about 430-365 B.C.), of an Elements
of Arithmetic in the form which we call Euclidean ; and no
doubt text-books of the sort existed even before Archytas,
which probably Archytas himself and others after him im-
proved and developed in their turn.
The 'irrational'.
We mentioned above the dictum of Proclus (if the reading
dXoyoav is right) that Pythagoras discovered the theory, or
study, of irrationals. This subject was regarded by the
Greeks as belonging to geometry rather than arithmetic.
The irrationals in Euclid, Book X, are straight lines or areas,
and Proclus mentions as special topics in geometry matters
relating (1) to positions (for numbers have no position), (2) to
contacts (for tangency is between continuous things), and (3)
to irrational straight lines (for where there is division ad
infinitum, there also is the irrational). 4 I shall therefore
postpone to Chapter V on the Pythagorean geometry the
question of the date of the discovery of the theory of irra-
tionals. But it is certain that the incommensurability of the
1 Nicom. ii. 24. 6, 7.
2 Musici Scriptores Graeci, ed. Jan, pp. 148-66; Euclid, vol. viii, ed.
Heiberg and Menge, p. 162.
3 Boetius, De Inst. Musica, iii. 11 (pp. 285-6, ed. Friedlein) ; see Biblio-
theca Mathematica, vi 3 , 1905/6, p. 227.
4 Proclus on Eucl. I, p. 60. 12-16.
THE 'IRRATIONAL' 91
diagonal of a square with its side, that is, the ' irrationality '
of V2, was discovered in the school of Pythagoras, and it is
more appropriate to deal with this particular case here, both
because the traditional proof of the fact depends on the
elementary theory of numbers, and because the Pythagoreans
invented a method of obtaining an infinite series of arith-
metical ratios approaching more and more closely to the value
of y/2.
The actual method by which the Pythagoreans proved the
fact that V2 is incommensurable with 1 was doubtless that
indicated by Aristotle, a reductio ad absurdum showing that,
if the diagonal of a square is commensurable with its side, it
will follow that the same number is both odd and even. 1 This
is evidently the proof interpolated in the texts of Euclid as
X. 117, which is in substance as follows :
Suppose AG, the diagonal of a square, to be commensur-
able with AB, its side ; let a : fi be their ratio expressed in
the smallest possible numbers.
Then oc >/?, and therefore oc is necessarily > 1.
Now AC 2 :AB 2 = ot 2 :/3 2 ; '
and, since AC 2 = 2 AB 2 , ex 2 = 2 /3 2 .
Hence a 2 , and therefore oc, is even.
Since a : /? is in its lowest terms, it follows that (3 must
be odd.
Let oc = 2 y ; therefore 4 y 2 = 2 /3 2 , or 2 y 2 = /3 2 , so that 2 ,
and therefore /?, is even.
But /3 was also odd : which is impossible.
Therefore the diagonal AC cannot be commensurable with
the side AB.
Algebraic equations.
(a) ' Side- and 'diameter-' numbers, giving successive
approximations to V 2.
The Pythagorean method of finding any number of succes-
sive approximations to the value of V2 amounts to finding
all the integral solutions of the indeterminate equations
2x 2 -y 2 = +1,
the solutions being successive pairs of what were called side-
1 Arist. Anal. pr. i. 23, 41 a 26-7.
92 PYTHAGOREAN ARITHMETIC
and diameter- (diagonal-) numbers respectively. The law of
formation of these numbers is explained by Theon of Smyrna,
and is as follows. 1 The unit, being the beginning of all things,
must be potentially both a side and a diameter. Consequently
we begin with two units, the one being the first side, which we
will call a x , the other being the first diameter, which we will
call d r
The second side and diameter (a 2 , d 2 ) are formed from the
first, the third side and diameter (a 3 , d 3 ) from the second, and
so on, as follows :
a 2 = a x + d x , d 2 = 2a l + d 1 ,
a 3 = a 2 + d 2 , d ?> = 2 a 2 + d 2 ,
a n+l — U n -\-d n , ( t n +\ — ^ ®n~y™n m
Since a 1 = d 1 = 1, it follows that
a 2 = 1 + 1 = 2,
c? 2 = 2. 1 + 1 =. 3,
a 3 = 2 + 3 = 5,
d 3 = 2 . 2 + 3 = 7,
a± = 5 + 7 = 12,
d 4 = 2 . 5 + 7 =17,
and so on.
Theon states, with reference to these numbers, the general
proposition that
d n * = 2a*±l,
and he observes (1) that the signs alternate as successive d'f*
and a' a are taken, d l 2 — 2a 1 2 being equal to —\,d 2 2 — 2a 2
equal to + 1, d 3 2 — 2a 3 2 equal to —1, and so on, while (2) the
sum of the squares of all the d's will be double of the squares
of all the a's. [If the number of successive terms in each
series is finite, it is of course necessary that the number should
be even.]
The properties stated depend on the truth of the following-
identity
{2x + y) 2 -2(x + y) 2 = 2x 2 -y 2 ;
for, if x, y be numbers which satisfy one of the two equations
2x 2 — y 2 = + 1 ,
the formula (if true) gives us two higher numbers, x + y and
2 x + y, which satisfy the other of the two equations.
Not only is the identity true, but we know from Proclus
1 Theon of Smyrna, pp. 43, 44.
'SIDE-' AND 'DIAMETER-' NUMBERS 93
how it was proved. 1 Observing that ' it is proved by him
(Euclid) graphically (ypa/x/iLKco?) in the Second Book of the
Elements'. Proclus adds the enunciation of Eucl. II. 10.
This proposition proves that, if A B is bisected at G and pro-
duced to D, then
AD 2 + DB 2 = 2AG 2 + 2CD 2 ;
and, if AC — GB — x and BD = y, this gives
(2x + y) 2 + y 2 = 2x 2 + 2(x + y) 2 ,
or (203 + yf - 2 (x + yf = 2 x 2 - y 2 ,
*
which is the formula required.
We can of course prove the property of consecutive side-
and diameter- numbers algebraically thus :
d n 2 -2a n 2 = (2a n _ 1 + d n _ 1 ) 2 -2(a n _ 1 + d n _ 1 ) 2
— 9/7 2 _/7 2
= — (d n _ l -2a M _ 1 ")
= + {d n ^ 2 — 2 « n _ 2 2 )> i* 1 ^ e m & nn er ;
and so on.
In the famous passage of the Republic (546 c) dealing with
the geometrical number Plato distinguishes between the
'irrational diameter of 5 ', i.e. the diagonal of a square having
5 for its side, or \/(50), and what he calls the 'rational
diameter ' of 5. The square of the ' rational diameter ' is less
by 1 than the square of the ' irrational diameter ', and is there-
fore 49, so that the 'rational diameter 'is 7; that is, Plato
refers to the fact that 2 . 5 2 — 7 2 = 1, and he has in mind the
particular pair of side- and diameter- numbers, 5 and 7, which
must therefore have been known before his time. As the proof
of the property of these numbers in general is found, as Proclus
says, in the geometrical theorem of Eucl. II. 10, it is a fair
inference that that theorem is Pythagorean, and was prob-
ably invented for the special purpose.
1 Proclus, Comm. on Bep. of Plato, ed. Kroll, vol. ii, 1901, cc. 23 and
27, pp. 24, 25, and 27-9.
94 PYTHAGOREAN ARITHMETIC
(/3) The €7rdi/$7]fia (' bloom ') of Thymaridas.
Thymaridas of Paros, an ancient Pythagorean already
mentioned (p. 69), was the author of a rule for solving a
certain set of n simultaneous simple equations connecting n
unknown quantities. The rule was evidently well known, for
it was called by the special name of k-rrdvO-qpLa, the ' flower ' or
' bloom ' of Thymaridas. 1 (The term kirdvO-qpa is not, how-
ever, confined to the particular proposition now in question ;
Iamblichus speaks of kiravBripara of the Introductio arith-
Tnetica, 'arithmetical kiravB-qpaja' and kiravQr]pLara of par-
ticular numbers.) The rule is stated in general terms and no
symbols are used, but the content is pure algebra. The known
or determined quantities ((hpicrpkuou) are distinguished from
the undetermined or unknown (dopiarov), the term for the
latter being the very word used by Diophantus in the expres-
sion nXfjdos /xoudScou dopio-Toi/, * an undefined or undetermined
number of units', by which he describes his dpi6p.6$ or un-
known quantity (= x). The rule is very obscurely worded,
but it states in effect that, if we have the following n equa-
tions connecting n unknown quantities x, x ly x 2 . . . x n _ 1}
namely
X ~r X^ -J- 3?2 ~r • • • i 3? w _i — - S,
X -p X-i — Qj-i ,
X "T" Xo — wo
x + x n _ l = a n _ 1 ,
the solution is given by
_ (a^+ a 2 + .. . + c^ -j) — s
x — - — ■ •
n— 2
Iamblichus, our informant on this subject, goes on to show
that other types of equations can be reduced to this, so that
the rule does not ( leave us in the lurch ' in those cases either. 2
He gives as an instance the indeterminate problem represented
by the following three linear equations between four unknown
quantities :
x + y = a (z + u),
x+ z = b(u + y),
x + u= c(y + z).
1 Iambi, in Nicom., p. 62. 18 sq. 2 lb., p. 63. 16.
THE 'EriAN0HMA ('BLOOM') OF THYMARIDAS 95
From these equations we obtain
x+y+z+w = (a+l)(z + u) = (6+1) (u + y) = (c+ 1) (y + z).
If now x, y, z, u are all to be integers, x + y 4- z + u must
contain a + l,6+l,c+l as factors. If L be the least common
multiple of a + 1, b + 1, c+1, we can put £ + 2/ + 2 + w = Z, and
we obtain from the above equations in pairs
a ,
* C6+1
x + z = j-— L,
6+1
C T
x + w= L,
c + 1
while x + y + z + u = L.
These equations are of the type to which Thymaridas's rule
applies, and, since the number of unknown quantities (and
equations) is 4, n — 2 is in this case 2, and
r / a b c \
/ 'l a+ l + r + l + c + l,>- i
X = 2
The numerator is integral, but it may be an odd number, in
which case, in order that x may be integral, we must take 2 L
instead of L as the value of x + y + z + u.
Iamblichus has the particular case where a= 2,b = 3,c= 4.
L is thus 3.4.5 = 60, and the numerator of the expression for
x becomes 133 — 60, or 73, an odd number; he has therefore
to put 2L or 120 in place of L, and so obtains x= 73, y— 7,
0=17, u = 23.
Iamblichus goes on to apply the method to the equations
4
x + z = -(u + y),
x + u = ~(y + z),
96 PYTHAGOREAN ARITHMETIC
which give
579
x+y+z+u = ~(z + u)=-(u + y) - ~(y + z).
Therefore
x + y + z + u = - (x + y) = - (x + z) = - (x + u,).
O TC O
In this case we take Z, the least common multiple of 5, 7, 9,
or 315, and put
x + y + z + u = L — 315,
x + y = -L — 189,
o
^ + 2; = -L = 180,
7
ic + u= -L = 175,
9
, 544-315 229
whence x = — = — •
2 2
In order that x may be integral, we have to take 2L, or 630,
instead of L, or 315, and the solution is x — 229, y — 149,
z — 131, u = 121.
(y) J.?m o/ rectangles in relation to perimeter.
Sluse, 1 in letters to Huygens dated Oct. 4, 1657, and Oct. 25,
1658, alludes to a property of the numbers 16 and 18 of
which he had read somewhere in Plutarch that it was known
to the Pythagoreans, namely that each of these numbers
represents the perimeter as well as the area of a rectangle ;
for 4. 4 = 2. 4 + 2. 4 and 3.6 = 2.3 + 2.6. I have not found the
passage of Plutarch, but the property of 16 is mentioned in the
Theologumena Arithmetic >es, where it is said that 16 is the only
square the area of which is equal to its perimeter, the peri-
meter of smaller squares being greater, and that of all larger
squares being less, than the area. 2 We do not know whether
the Pythagoreans proved that 16 and 18 were the only num-
bers having the property in question ; but it is likely enough
that they did, for the proof amounts to finding the integral
1 (Euvres completes de C. Huygens, pp. 64, 260.
2 Theol At:, pp. 10, 23 (Ast).
>*
TREATISES ON ARITHMETIC 97
solutions of xy = 2 (x + y). This is easy, for the equation is
equivalent to (x—2) (y—2) = 4, and we have only to equate
x — 2 and y — 2 to the respective factors of 4. Since 4 is only
divisible into integral factors in two ways, as 2 . 2 or as 1 . 4,
we get, as the only possible solutions for x, y, (4, 4) or (3, 6).
Systematic treatises on arithmetic (theory of
numbers).
It will be convenient to include in this chapter some
account of the arithmetic of the later Pythagoreans, begin-
ning with Nicomachus. If any systematic treatises on
arithmetic were ever written between Euclid (Books VII-IX)
and Nicomachus, none have survived. Nicomachus, of
Gerasa, probably the Gerasa in Judaea east of the river
Jordan, flourished about 100 A.D., for, on the one hand, in
a work of his entitled the Enchiridion Harmonices there is
an allusion to Thrasyllus, who arranged the Platonic dialogues,
wrote on music, and was the astrologer-friend of Tiberius ; on
the other hand, the Introductio Arithmetica of Nicomachus
was translated into Latin by Apuleius of Madaura under the
Antonines. Besides the 'Api6fj,T]TiKr) elcrayooyrj, Nicomachus
is said to have written another treatise on the theology or the
mystic properties of numbers, called ©eo\oyovfj.eva dpidur]-
TiKrjs, in two Books. The curious farrago which has come
down to us under that title and which was edited by Ast 1 is,
however, certainly not by Nicomachus ; for among the authors
from whom it gives extracts is Anatolius, Bishop of Laoclicaea
(a.d. 270); but it contains quotations from Nicomachus which
appear to come from the genuine work. It is possible that
Nicomachus also wrote an Introduction to Geometry, since in
one place he says, with regard to certain solid numbers, that
they have been specially treated ' in the geometrical intro-
duction, being more appropriate to the theory of magnitude ' 2 ;
but this geometrical introduction may not necessarily have
been a work of his own.
It is a very far cry from Euclid to Nicomachus. In the
1 Theologumena arithmeticae. Accedit Nicomachi Geraseni Institutio
arithmetica, ed. Ast, Leipzig, 1817.
2 Nicom. Ariihm. ii. 6. 1.
1523 H
98 PYTHAGOREAN ARITHMETIC
Introcluctio arithmetica we find the form of exposition
entirely changed. Numbers are represented in Euclid by
straight lines with letters attached, a system which has the
advantage that, as in algebraical notation, we can work with
numbers in general without the necessity of giving them
specific values ; in Nicomachus numbers are no longer de-
noted by straight lines, so that, when different undetermined
numbers have to be distinguished, this has to be done by
circumlocution, which makes the propositions cumbrous and
hard to follow, and it i is necessary, after each proposition
has been stated, to illustrate it by examples in concrete
numbers. Further, there are no longer any proofs in the
proper sense of the word ; when a general proposition has been
enunciated, Nicomachus regards it as sufficient to show that
it is true in particular instances ; sometimes we are left to
infer the general proposition by induction from particular
cases which are alone given. Occasionally the author makes
a quite absurd remark through failure to distinguish between
the general and the particular case, as when, after he has
defined the mean which is ' subcontrary to the harmonic ' as
being determined by the relation 7 = -, where a>b>c?
and has given 6, 5, 3 as an illustration, he goes on to observe
that it is a property peculiar to this mean that the product of
the greatest and middle terms is double of the product of the
middle and least, 1 simply because this happens to be true in
the particular case ! Probably Nicomachus, who was not
really a mathematician, intended his Introduction to be, not
a scientific treatise, but a popular treatment of the subject
calculated to awaken in the beginner an interest in the theory
of numbers by making him acquainted with the most note-
worthy results obtained up to date ; for proofs of most of his
propositions he could refer to Euclid and doubtless to other
treatises now lost. The style of the book confirms this hypo-
thesis ; it is rhetorical and highly coloured ; the properties of
numbers are made to appear marvellous and even miraculous ;
the most obvious relations between them are stated in turgid
language very tiresome to read. It was the mystic rather
than the mathematical side of the theory of numbers that
1 Nicom. ii. 28. 3.
NICOMACHUS 99
interested Nicomachus. If the verbiage is eliminated, the
mathematical content can be stated in quite a small com-
pass. Little or nothing in the book is original, and, except
for certain definitions and refinements of classification, the
essence of it evidently goes back to the early Pythagoreans.
Its success is difficult to explain except on the hypothesis that
it was at first read by philosophers rather than mathemati-
cians (Pappus evidently despised it), and afterwards became
generally popular at a time when there were no mathemati-
cians left, but only philosophers who incidentally took an
interest in mathematics. But a success it undoubtedly was ;
this is proved by the number of versions or commentaries
which appeared in ancient times. Besides the Latin transla-
tion by Apuleius of Madaura (born about A.D. 125), of which
no trace remains, there was the version of Boetius (born about
480, died 524 A.D.); and the commentators include Iamblichus
(fourth century), Heronas, 1 Asclepius of Tralles (sixth century),
Joannes Philoponus, Proclus. 2 The commentary of Iamblichus
has been published, 3 as also that of Philoponus, 4 while that of
Asclepius is said to be extant in MSS. When (the pseudo-)
Lucian in his Philoimtris (c. 12) makes Critias say to Triephon
' you calculate like Nicomachus ', we have an indication that
the book was well known, although the remark may be less a
compliment than a laugh at Pythagorean subtleties. 5
Book I of the Introductio, after a philosophical prelude
(cc. 1-6), consists principally of definitions and laws of forma-
tion. Numbers, odd and even, are first dealt with (c. 7); then
comes the subdivision of even into three kinds (1) evenly-even,
of the form 2 n , (2) even-odd, of the form 2 (2 n + 1), and (3)
odd-even, of the form 2 m+1 (2n + 1), the last-named occupying
a sort of intermediate position in that it partakes of the
character of both the others. The odd is next divided into
three kinds : (1) 'prime and incomposite ', (2) ' secondary and
1 v. Eutoc. in Archim. (ed. Heib. iii, p. 120. 22). 2 v. Suidas.
3 The latest edition is Pistelli's (Teubner, 1894).
4 Ed. Hoche, Heft 1, Leipzig, 1864, Heft 2, Berlin, 1867.
5 Triephon tells Critias to swear by the Trinity ('One (proceeding) from
Three and Three from One '), and Critias replies, ' You would have me
learn to calculate, for your oath is mere arithmetic and you calculate
like Nicomachus of Gerasa. I do not know what you mean by your
" One-Three and Three-One " ; I suppose you don't mean the rerpafcr^
of Pythagoras or the oydods or the rpiaKas ? '
H 2
100 PYTHAGOREAN ARITHMETIC
composite ', a product of prime factors (excluding 2, which is
even and not regarded as prime), and (3) ' that which is in itself
secondary and composite but in relation to another is prime and
incomposite ', e.g. 9 in relation to 25, which again is a sort of
intermediate class between the two others (cc. 11-13); the
defects of this classification have already been noted (pp. 73-4).
In c. 13 we have these different classes of odd numbers ex-
hibited in a description of Eratosthenes's ' sieve ' (koo-klvop), an
appropriately named device for finding prime numbers. The
method is this. We set out the series of odd numbers begin-
ning from 3.
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31,
Now 3 is a prime number, but multiples of 3 are not ; these
multiples, 9, 15... are got by passing over two numbers at
a time beginning from 3 ; we therefore strike out these num-
bers as not being prime. Similarly 5 is a prime number, but
by passing over four numbers at a time, beginning from 5, we
get multiples of 5, namely 15, 25 ... ; we accordingly strike
out all these multiples of 5. In general, if n be a prime num-
ber, its multiples appearing in the series are found by passing
over n—l terms at a time, beginning from n ; and we can
strike out all these multiples. When we have gone far enough
with this process, the numbers which are still left will be
primes. Clearly, however, in order to make sure that the
odd number 2 n + 1 in the series is prime, we should have to
try all the prime divisors between 3 and V{2n+1)\ it is
obvious, therefore, that this primitive empirical method would
be hopeless as a practical means of obtaining prime numbers
of any considerable size.
The same c. 13 contains the rule for finding whether two
given numbers are prime to one another ; it is the method of
Eucl. VII. 1, equivalent to our rule for finding the greatest
common measure, but Nicomachus expresses the whole thing
in words, making no use of any straight lines or symbols to
represent the numbers. If there is a common measure greater
than unity, the process gives it ; if there is none, i. e. if 1 is
left as the last remainder, the numbers are prime to one
another.
chapters (cc. 14-16) are on over-perfect (v7re preXrjs).
IBHMiH
NICOMACHUS
101
deficient (kWnrfis), and perfect (reAeioy) numbers respectively.
The definitions, the law of formation of perfect numbers,
and Nicomachus's observations thereon have been given above
(P- 74).
Next comes (cc. 17-23) the elaborate classification of
numerical ratios greater than unity, with their counterparts
which are less than unity. There are five categories of each,
and under each category there is (a) the general name, (6) the
particular names corresponding to the particular numbers
taken.
The enumeration is tedious, but, for purposes of reference,
is given in the following table : —
RATIOS GREATER THAN UNITY
RATIOS LESS THAN UNITY
1. (a) General
1. (a) General
7roXAa7rAa(rtos,
(multiplex)
multiple
V7ro7ro\\a7r\d(Tto<s,
(submultiplex)
submultiple
(b) Particular
(b) Particular
$L7r\d<rtos,
double
V7TO§LTr\d(TlO<i,
one half
(duplus)
Tpiir\d(TLO<S,
(triplus)
&c.
triple
(subduplus)
VTrOTpiTrXdiTLOS)
(subtriplus)
&c.
one third
2. (a) General
2. (a) General
€7Tt/xdptOS
, r r , . , • \ f a num-
(superparticulans) J
ber which is of the form
1 n+1
1 + or ,
n n
where n is any integer.
v7re7ri/xdptos (subsuper- ) ,
particulars) j e
fraction 7 , where n is
n + 1 1
any integer.
(b) Particular
(b) Particular
According to the value of
n, we have the names
rjfjLioXios = 1^
(sesquialter)
€7rtT/)tros) = l^j
(sesquitertius)
C7TtTeTapTOS = 1^
(sesquiquartus)
&c.
vcfirjiALoXios
(subsesquialter)
V7T€7rtTpiT09
(subsesquitertius)
V7T€7rtTeTapros
(subsesquiquartus
&c.
2
— 3
3
~~ 4
4
— 5
102
PYTHAGOREAN ARITHMETIC
RATIOS GREATER THAN UNITY
RATIOS LESS THAN UNITY
3. (a) General
3. (a) General
few* l whichex .
(superpartiens) j
(subsuperpartiens) )
ceeds 1 by twice, thrice,
... „ m + n
or more times a sub-
ot the form ~ •
2m + w
multiple, and which
therefore may be repre-
sented by
m 2m + n
If nv
1 "T i 01 i
m-f-w m + n
(b) Particular
The formation of the names
for the series of particular super-
t
partientes follows three different
plans.
#
Thus, of numbers of the form
+ m + l'
( €7TlSl//,e/3^S
The corresponding names are
(superbipartiens)
not specified in Nicomachus.
1 ~k 1 or e7riSiVpiTOS
(superbitertius)
V or 8i<re7rtT/HTOS
/ £7riTpL/JL€pyj<S
(supertripartiens)
< L 4
or €7rtT/3iTerapros
(super triquartus)
V 01* Tpt(T€7riTiTapTOS
I i7rLT€TpafX€pr]<;
(superquad ripartiens)
1^ ]S n , 01* €7rn-€T/3a7r€/X7TTOS
(superquadriquintus)
^ or T€TpaKio-e7ri7re/x7TTOS
&c.
As regards the first name in
each case we note that, with
i7ri8ipi€prjs we must understand
TpiTiDV ) Wltll llTLTpi[X€pr)^, T€TCtp-
to>v, anc
so on.
NICOMACHUS
103
RATIOS GREATER THAN UNITY
Where the more general form
1 +
m
, instead of 1 +
m
m + n 7 ' ' m + l'
has to be expressed, Nicoma-
chus uses terms following the
third plan of formation above,
e.g.
1|- = T/)i(TC7ri7re/>t7rT09
1-^ = T€T/3a/a(re<£e/?So//,os
1|- = 7T€VTa/acre7reVaTOS
and so on, although he might
have used the second and called
these ratios €7rn-pi7r€/x7rTos 3 &c.
4. (a) General
TroWaTr\a(TL€.Tri[x6pio<;
(multiplex superparticularis)
This contains a certain mul-
tiple plus a certain submultiple
(instead of 1 plus a submultiple)
and is therefore of the form
m + - (instead of the 1 + - of
n n
the €7ri/xo/3ios) or
mn + 1
n
(b) Particular
2-g- = 8i7rAa(rie<£^//,icn;s
(duplex sesquialter)
2^ = 8i7rAaa"te7rtrptTos
(duplex sesquitertius)
3-|- = T/3i7rAaa'ie7U7re/A7rTOS
(triplex sesquiquintus)
&c.
5. (a) General
7roWa7r\acTL€TrifAepy$
(multiplex superpartiens).
This is related to cVi/xep^s
[(3) above] in the same way as
7roAAa7rAacri€7n/xdpios to €7ri/xopios ;
that is to say, it is of the form
p +
m
m + n
or
(p + l)m + n
m + n
RATIOS LESS THAN UNITY
<£. (a) General
v7ro7roWa.7r\acrLe7rL[Aopios
(submultiplex superparticularis)
n
of the form
mn + 1
The corresponding particular
names do not seem to occur in
Nicomachus, but Boetius has
them, e. g. subduplex sesquialter,
subduplex sesquiquartus.
5. (a) General
v7ro7roAAa.7rAao-ie7ri/xep^s
(subm ultiplex superpartiens),
a fraction of the form
m+n
(p+l)m+n
104
PYTHAGOREAN ARITHMETIC
RATIOS GREATER THAN UNITY
(b) Particular
These names are only given
for cases where n = 1 j thej''
follow the first form of the
names for particular cVt/^epei?,
e.g.
2f = 8i7rAacrt€7rt8i/Aep^s
(duplex superbipartiens)
&c.
RATIOS LESS THAN UNITY
Corresponding names not
found in Nicomachus ; but
Boetius has sub duplex super-
bipartiens, &c.
In c. 23 Nicomachus shows how these various ratios can be
got from one another by means of a certain rule. Suppose
that
a, b, c
are three numbers such that a:b = b\c = one of the ratios
described ; we form the three numbers
a,
a+b a+2b+c
and also the three numbers
c, c + b, c + 2b + a
Two illustrations may be given. If a = b = c = 1, repeated
application of the first formula gives (1, 2, 4), then (1, 3, 9),
then (1, 4, 16), and so on, showing the successive multiples.
Applying the second formula to (1, 2, 4), we get (4, 6, 9) where
the ratio is § ; similarly from (1, 3, 9) we get (9, 12, 16) where
the ratio is § , and so on ; that is, from the TroWanXacrioi we
get the kiTLfiopLoi. Again from (9, 6, 4), where the ratio is
of the latter kind, we get by the first formula (9, 15, 25),
giving the ratio If, an e7njjL€prj?, and by the second formula
(4, 10, 25), giving the ratio 2i, &7ro\\a7r\a(rie7ri/i6pios. And
so on.
Book II begins with two chapters showing how, by a con-
verse process, three terms in continued proportion with any
one of the above forms as common ratio can be reduced to
three equal terms. If
a, b, c
NICOMACHUS 105
are the original terms, a being the smallest, we take three
terms of the form
a, b — a, {c — a — 2(b — a)} = c + a — 2b,
then apply the same rule to these three/and so on.
In cc. 3-4 it is pointed out that, if
1, r, r 2 ..., r n ...
be a geometrical progression, and if
an kmiiopios ratio,
n — /y.71 — 1 i /viW
rn ' ' ' j
then
Pn z= r + 1
and similarly, if
P'n= Pn-1 + Pn>
Pn' * '
and so on.
If we set out in rows numbers formed in this way,
I r, r 2 , ? i3 ... r w
r+1, r 2 + r, r 3 + r 2 ... r n + r n-i
r* + 2r+l, r 3 + 2r 2 + r... r n + 2 r n_1 + r n ~ 2
r 3 + 3r 2 + 3^4.1,,. r w + 3r n-i + 3r n-2 + r w-3
T n + nr n-l + ^V 71 " 1 ) r n-2 + <t> +1
the vertical rows are successive numbers in the ratio r/(r + 1),
while diagonally we have the geometrical series 1, r+1,
(r+1) 2 , (r+1) 3 ....
Next follows the theory of polygonal numbers. It is pre-
faced by an explanation of the quasi-geometrical way of
representing numbers by means of dots or a's. Any number
from 2 onwards can be represented as a line ; the plane num-
bers begin with 3, which is the first number that can be
represented in the form of a triangle ; after triangles follow
squares, pentagons, hexagons, &c. (c. 7). Triangles (c. 8) arise
by adding any number of successive terms, beginning with 1,
of the series of natural numbers
1,2, 3, ... n, ....
106 PYTHAGOREAN ARITHMETIC
The gnomons of triangles are therefore the successive natural
numbers. Squares (c. 9) are obtained by adding any number
of successive terms of the series of odd numbers, beginning
with 1, or
1, 3, 5, ... 271—1,....
The gnomons of squares are the successive odd numbers.
Similarly the gnomons of pentagonal numbers (c. 10) are the
numbers forming an arithmetical progression with 3 as com-
mon difference, or
1,4,7,... l + (n-l)3,...;
and generally (c. II) the gnomons of polygonal numbers of a
sides are
1, l + (a-2), l+2(a-2),... l+(r-l)(a-2),...
and the a-gonal number with side n is
1 + 1 + (a - 2) + 1 + 2 (a - 2) + . . . + 1 + (n - 1 ) (a - 2)
= n + \n (n- 1) (a — 2)
The general formula is not given by Nicomachus, who con-
tents himself with writing down a certain number of poly-
gonal numbers of each species up to heptagons.
After mentioning (c. 12) that any square is the sum of two
successive triangular numbers, i.e.
n 2 = J (n - 1) n + \ n (n + 1),
and that an a-gonal number of side n is the sum of an
(a— l)-gonal number of side n plus a triangular number of
side n — 1 , i. e.
n + ±n(n — I) (a — 2) — n + \n(n- l) (a-S) + ^n (n- 1),
he passes (c. 13) to the first solid number, the pyramid. The
base of the pyramid may be a triangular, a square, or any
polygonal number. If the base has the side n, the pyramid is
formed by similar and similarly situated polygons placed
successively upon it, each of which has 1 less in its side than
that which precedes it ; it ends of course in a unit at the top,
the unit being ' potentially ' any polygonal number. Nico-
machus mentions the first triangular pyramids as being 1, 4,
10, 20, 35, 56, 84, and (c. 14) explains the formation of the
series of pyramids with square bases, but he gives no general
NICOMACHUS 107
formula or summation. An <x-gonal number with n in its
side being
n + ^n(n--l) (a — 2),
it follows that the pyramid with that polygonal number for
base is
l + 2 + 3 + ...+?i + §(a-2) {1.2 + 2. 3 + ...+(71-1)™}
n(n+l) a — 2 (n — l)n{n+l)
— _[. . .
2 2 3
A pyramid is KoXovpos, truncated, when the unit is cut off
the top, SiKoXovpos, twice-truncated, when the unit and the
next layer is cut off, rpiKoXovpos, thrice-truncated, when three
layers are cut off, and so on (c. 14).
Other solid numbers are then classified (cc. 15-17): cubes,
which are the product of three equal numbers ; scalene num-
bers, which are the product of three numbers all unequal,
and which are alternatively called wedges (o-epwvio-Koi), stakes
(crept) Kio-Koi), or altars (ftcofiicrKoi). The latter three names are
in reality inappropriate to mere products of three unequal
factors, since the figure which could properly be called by
these names should taper, i.e. should have the plane face at
the top less than the base. We shall find when we come to
the chapter on Heron's mensuration that true (geometrical)
ficofxicTKOL and cr(prjvi<rKOL have there to be measured in which
the top rectangular face is in fact smaller than the rectangular
base parallel to it. Iamblichus too indicates the true nature
of ftcofjiicrKoi and ct^v'kjkoi when he says that they have not
only their dimensions but also their faces and angles unequal,
and that, while the ttXlvOl? or Sokis corresponds to the paral-
lelogram, the cr<fir)vicrKos corresponds to the trapezium. 1 The
use, therefore, of the terms in question as alternatives to scalene
appears to be due to a misapprehension. Other varieties of
solid numbers are parallelepipeds, in which there are faces
which are erepoyu^eiy (oblong) or of the form n(n+l), so
that two factors differ by unity ; beams (SoKide?) or columns
(crTTjXiSes, Iamblichus) of the form m 2 (m + n); tiles (ttXlvOlSzs)
of the form m 2 (m — n). Cubes, the last digit (the units) of
which are the same as the last digit in the side, are spherical
1 Iambi, in Nicom., p. 93. 18, 94. 1-3.
108 PYTHAGOREAN ARITHMETIC
(o-(f)cupiKoi) or recurring (anoKaTaa-raTLKoi) ; these sides and
cubes end in 1,5, or 6, and, as the squares end in the same
digits, the squares are called circular (kvkXlkol).
Oblong numbers (eTepo/MrJKeis) are, as we have seen, of the
form m(m+l); prolate numbers (TrpouiJKeis) of the form
m (m + n) where n > 1 (c. 1 8). Some simple relations between
oblong numbers, squares, and triangular numbers are given
(cc. 19-20). If h n represents the oblong number n (n+. I), and
t n the triangular number ^n(n+l) of side n, we have, for
example,
K A 2 — ( n + !)A K - n2 = n > n VK-\ = n/(n- 1),
n 2 /h n =h n /(n + l) 2 , n 2 + (n + l) 2 + 2h n = (2n+l) 2 ,
n 2 + h n = t 2ni h n + (n+l) 2 = t 2n+1 ,
n 2 + n= J 1 * I,
all of which formulae are easily verified.
Sum of series of cube numbers.
C. 20 ends with an interesting statement about cubes. If,
says Nicomachus, we set out the series of odd numbers
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
the first (1) is a cube, the sum of the next two (3 + 5) is a
cube, the sum of the next three (7 + 9 + 11) is a cube, and so on.
We can prove this law by assuming that n z is equal to the
sum of n odd numbers beginning with 2x+l and ending
with 2x + 2n— 1. The sum is (2x + n)n; since therefore
(2x-\-n)n = n 3 ,
x = |- (n 2 — n),
and the formula is
(n 2 — n+ 1) + (h 2 — n + 3) + ... +(n 2 + n— 1) = n 3 .
By putting successively n — 1,2,3 ... r, &c, in this formula
and adding the results we find that
l 3 + 2 3 + 3 3 +... +r 3 = 1+ (3 + 5) +(7 + 9 + 11)+... + (... r 2 + r- 1).
The number of terms in this series of odd numbers is clearly
1+2 + 3 + .. .+r or |r(r+l).
Therefore l 3 + 2 3 + 3 3 + . . . + r 3 = i r (r + 1) (1 + r 2 + r - 1)
= {ir(r+l)} 2 .
SUM OF SERIES OF CUBE NUMBERS 109
Nicomachus does not give this formula, but it was known
to the Roman agrimensores, and it would be strange if
Nicomachus was not aware of it. It may have been dis-
covered by the same mathematician who found out the
proposition actually stated by Nicomachus, which probably
belongs to a much earlier time. For the Greeks were from
the time of the early Pythagoreans accustomed to summing
the series of odd numbers by placing 3, 5, 7, &c, successively
as gnomons round 1 ; they knew that the result, whatever
the number of gnomons, was always a square, and that, if the
number of gnomons added to 1 is (say) r, the sum (including
the 1) is (r+1) 2 . Hence, when it was once discovered that
the first cube after 1, i.e. 2 3 , is 3 + 5, the second, or 3 3 , is
7 + 9 + 11, the third, or 4 3 , is 13 + 15 + 17 + 19, and so on, they
were in a position to sum the series 1 3 + 2 3 + 3 3 + ... -fr 3 ;
for it was only necessary to find out how many terms of the
series 1 + 3 + 5 + . . . this sum* of cubes includes. The number
of terms being clearly 1 + 2 + 3 + . . . + r, the number of
gnomons (including the 1 itself) is %r(r + l); hence the sum
of them all (including the 1), which is equal to
l 3 + 2 3 + 3 3 +...+r 3 ,
is {-|r(r+l)} 2 . Fortunately we possess apiece of evidence
which makes it highly probable that the Greeks actually
dealt with the problem in this way. Alkarkhi, the Arabian
algebraist of the tenth-eleventh century, wrote an algebra
under the title Al-Fakhri. It would seem that there were at
the time two schools in Arabia which were opposed to one
another in that one favoured Greek, and the other Indian,
methods. Alkarkhi was one of those who followed Greek
models almost exclusively, and he has a proof of the theorem
now in question by means of a figure with gnomons drawn
in it, furnishing an excellent example of the geometrical
algebra which is so distinctively Greek.
Let AB be the side of a square AG; let
AB = l + 2 + ...+n = %n(n+l),
and suppose BB' = n, B f B" = n—1, B" B'" — n — 2, and so on.
Draw the squares on AB', AB" ... forming the gnomons
shown in the figure.
no
PYTHAGOREAN ARITHMETIC
B_ C
B'
B"
QUI
C'
D"
D' D
Then the gnomon
BCD = BBf . BC + DD'. C ' D'
= BB'(BC+C'D').
Now BG = ±n(n + l),
C'D'=l+2 + 3 + ... + (n-l) = ±n(n-l), BB'=n;
therefore (gnomon BC D) — n .n 2 = n z .
Similarly (gnomon B' C" D') = (n — l) 3 , and so on.
Therefore l 3 + 2 3 + ... +n z = the sum of the gnomons round
the small square at A which has 1 for its side plus that small
square ; that is,
l3 + 2 3 + 3 a +... + n. 3 = square AC = {±n(n+ l)p.
It is easy to see that the first gnomon about the small
square at A is 3 + 5 = 2 3 , the next gnomon is 7 + 9 + 11 = 3 3 ,
and so on.
The demonstration therefore hangs together with the
theorem stated by Nicomachus. Two alternatives are possible.
Alkarkhl may have devised the proof himself in the Greek
manner, following the hint supplied by Nicomachus's theorem.
Or he may have found the whole proof set out in some
Greek treatise now lost and reproduced it. Whichever alter-
native is the true one, we can hardly doubt the Greek origin
of the summation of the series of cubes.
Nicomachus passes to the theory of arithmetical proportion
and the various means (cc. 21-9), a description of which has
already been given (p. 87 above). There are a few more
propositions to be mentioned under this head. If a—b = b — c,
so that a, b, c are in arithmetical progression, then (c. 23. 6)
b 2 -ac = {a~bf = (b-c) 2 ,
NICOMACHUS 111
a fact which, according to Nicomachus, was not generally
known. Boetius 1 mentions this proposition which, if we
take a + d, a, a — d as the three terms in arithmetical pro-
gression, may be written a 2 = (a + d) (a—d) + d 2 . This is
presumably the origin of the regula Nicomachi quoted by
one Ocreatus (1 O'Creat), the author of a tract, Prologus in
Helceph, written in the twelfth or thirteenth century 2
(' Helceph ' or ' Helcep ' is evidently equivalent to Algo-
rismus; may it perhaps be meant for the Al-Kafl of
Alkarkhi?). The object of the regula is to find the square
of a number containing a single digit. If d= 10 — a, or
a + d — 10, the rule is represented by the formula
a 2 = 10 (a-d) + d 2 ,
so that the calculation of a 2 is made to depend on that of d 2
which is easier to evaluate if d<a.
Again (c. 24. 3, 4), if a, b, c be three terms in descending
geometrical progression, r being the common ratio (a/b or b/c),
then
a — b_ a _ b
b — c be
and (a — b) = (r — 1)6, (b — c) — (r— l)c,
( a -b)-(b-c) = ( T -l)(b-c).
It follows that
b = a — b(r—\) — c + c (r— 1).
This is the property of three terms in geometrical pro-
gression which corresponds to the property of three terms
a, b, c of a harmonical progression
, a c
o = a = c + - ?
n n
from which we derive
n = (a + c) / (a—c).
If a, b, c are in descending order, Nicomachus observes
(c. 25) that 7 < = > - according as a, b, c are in arith-
v ' b c
metical, geometrical, or harmonical progression.
1 Boetius, Inst. Ar. ii. c. 43.
2 See Abh. zur Gesch. d. Math. 3, 1880, p. 134.
112 PYTHAGOREAN ARITHMETIC
The 'Platonic theorem' (c. 24. 6) about the number of
possible means (geometric) between two square numbers and
between two cube numbers respectively has already been
mentioned (pp. 89, 90), as also the 'most perfect proportion'
(p. 86).
Theon of Smyrna was the author of a book purporting
to be a manual of mathematical subjects such as a student
would require to enable him to understand Plato. A fuller
account of this work will be given later ; at present we are
only concerned with the arithmetical portion. This gives the
elementary theory of numbers on much the same lines as
we find it in Nicomachus, though less systematically. We
can here pass over the things which are common to Theon
and Nicomachus and confine ourselves to what is peculiar to
the former. The important things are two. One is the
theory of side- and diameter-numbers invented by the Pytha-
goreans for the purpose of finding the successive integral
solutions of the equations 2x 2 — y 2 =±l; as to this see
pp. 91-3 above. The other is an explanation of the limited
number of forms which square numbers may have. 1 If m 2 is
a square number, says Theon, either m 2 or m 2 — 1 is divisible
by 3, and again either m 2 or m 2 — 1 is divisible by 4 : which
is equivalent to saying that a square number cannot be of
any of the following forms, 3n + 2, 4w, -f- 2, 4w + 3. Again, he
says, for any square number m 2 , one of the following alterna-
tives must hold :
(1) , — both integral (e.g. m 2 = 4)>
TCv — 1 T)b
(2) , — both integral (e.g. m 2 — 9)»
(3) — , — both integral (e.g. m 2 = 36)»
<yyif 1 TYb 1
(4) , — both integral (e.g. m 2 = 25) •
1 Theon of Smyrna, p. 35. 17-36. 2.
ARITHMETIC IN THEON OF SMYRNA 113
lamblichus states the same facts in a slightly different form. 1
The truth of these statements can be seen in the following
way. 2 Since any number m must have one of the following
forms
6k, 6&+1, 6k±2, $k±3,
any square m 2 must have one or other of the forms
36/c 2 , 3Gk 2 ±12k+l, 36k 2 ±24k + 4 y 36k 2 ± 36fc + 9.
0Tb Oil
For squares of the first type — and — are both integral,
IYY\ 2 1 rwj 2 1
for those of the second type — - — > — - — are both integral,
o 4
/yyii J. 11% 2
for those of the third type -— - — and — are both integral,
0Tb TIl — 1
and for those of the fourth type — and are both
integral ; which agrees with Theon's statement. Again, if
the four forms of squares be divided by 3 or 4, the remainder
is always either or 1 ; so that, as Theon says, no square can
be of the form 3n + 2, 4n + 2, or 4^+3. We can hardly
doubt that these discoveries were also Pythagorean.
Iamblichus, born at Chalcis in Coele-Syria, was a pupil of
Anatolius and Porphyry, and belongs to the first half of the
fourth century A. D. He wrote nine Books on the Pythagorean
Sect, the titles of which were as follows : I. On the Life of
Pythagoras ; II. Exhortation to philosophy (TIpoTpeTrTLKos
km (biKocro^iav) ; III. On mathematical science in general ;
IV. On Nicomachus's Introductio Arithmetica ; V. On arith-
metical science in physics; VI. On arithmetical science in
ethics ; VII. On arithmetical science in theology ; VIII. On
the Pythagorean geometry ; IX. On the Pythagorean music.
The first four of these books survive and are accessible in
modern editions ; the other five are lost, though extracts
from VII. are doubtless contained in the Theologumena
arithmetices. Book IV. on Nicomachus's Introductio is that
which concerns us here ; and the few things requiring notice
are the following. The first is the view of a square number
1 Iambi, in Nicom., p. 90. 6-11.
2 Cf. Loria, Le scienze esatte nelV antica Grecia, p. 834.
1523 I
114 PYTHAGOREAN ARITHMETIC
as a race-course (8iav\o<?) : formed of successive numb ers
from 1 (as start, vo-irX-qg) up to n, the side of the square,
which is the turning-point (KafnrTrjp), and then back again
through (n— 1), (n — 2), &c, to 1 (the goal, vvcrcra), thus :
1+2 + 3 + 4... (n— 1).
n
1 + 2 + 3 + 4. ..(n-2) + (n-l) + .
This is of course equivalent to the proposition that n 2 is the
sum of the two triangular numbers \n (n + 1) and ^(n— 1) n
with sides n and n— 1 respectively. Similarly Iamblichus
points out 2 that the oblong number
7b(n—l) = (1 + 2 + 3+ ,..+n) + (n — 2 + n — 3 + ... + 3 + 2).
He observes that it was on this principle that, after 10,
which was called the unit of the second course (5eurepo)-
Sovfievr) fioj/ds), the Pythagoreans regarded 100 = 10.10 as
the unit of the third course (TpicoSovuevr) fiovds), 1000 — 10 3
as the unit of the fourth course (TeTpcoSovfieurj [lovcis), and
so on, 3 since
1 + 2 + 3 + ... + 10 + 9 + 8 + ...+2 + 1 = 10. 10,
10 + 20 + 30 + ... + 100 + 90 + 80+ ... + 20 + 10 = 10 3 ,
100 + 200 + 300 + ... + 1000 + 900 + ... + 200 + 100 = 10 4 ,
and so on. Iamblichus sees herein the special virtue of 10 :
but of course the same formulae would hold in any scale
of notation as well as the decimal.
In connexion with this Pythagorean decimal terminology
Iamblichus gives a proposition of the greatest interest. 4
Suppose we have any three consecutive numbers the greatest
of which is divisible by 3. Take the sum of the three
numbers ; this will consist of a certain number of units,
a certain number of tens, a certain number of hundreds, and
so on. Now take the units in the said sum as they are, then
as many units as there are tens in the sum, as many units as
there are hundreds, and so on, and add all the units so
obtained together (i.e. add the digits of the sum expressed
in our decimal notation). Apply the same procedure to the
J Iambi, in Nicom., p. 75. 25-77. 4. 2 lb., pp. 77. 4-80. 9.
3 lb., pp. 88. 15-90. 2. 4 lb., pp. 103. 10-104. 13.
IAMBLICHUS 115
result, and so on. Then, says Iamblichus, the final result
will be the number 6. E.g. take the numbers 10, 11, 12; the
sum is 33. Add the digits, and the result is 6. Take
994, 995, 996 : the sum is 2985 ; the sum of the digits is 24 ;
and the sum of the digits of 24 is again 6. The truth of the
general proposition is seen in this way. 1
Let Jf = w + 10w 1 + I0 2 n 2 + ...
be a number written in the decimal notation. Let S(jST)
represent the sum of its digits, S^ (JSf) the sum of the digits
of S(Jf) and so on.
Now N-S(N) == 9 (n 1 + ll.n i +.llln z + •••)>
whence Jf = 8 (If) (mod. 9).
Similarly 8(Jf) = S^N (mod. 9).
Let /S^" 1 ) (iV) = S&& (mod. 9)
be the last possible relation of this kind; S^Jf will be a
number JY T/ ^ 9.
Adding the congruences, we obtain
]f = N' (mod. 9), while N' < 9.
Now, if we have three consecutive numbers the greatest
of which is divisible by 3, we can put for their sum
N= (3^+l) + (3^ + 2) + (3p + 3) = 9p + 6,
and the above congruence becomes
9jp + 6 = N' (mod. 9),
so that lV = 6 (mod. 9) ;
and, since N f ^ 9, N' can only be equal to 6.
This addition of the digits of a number expressed in our
notation has an important parallel in a passage of the
Refutation of all Heresies by saint Hippolytus, 2 where there
is a description of a method of foretelling future events
called the ' Pythagorean calculus '. Those, he says, who
claim to predict events by means of calculations with numbers,
letters and names use the principle of the pythmen or base,
1 Loria, op. cit., pp. 841-2.
2 Hippolytus, Eefut. iv, c. 14.
I 2
116 PYTHAGOREAN ARITHMETIC
that is, what we call a digit of a number expressed in our
decimal notation ; for the Greeks, in the case of any number
above 9, the pythmen was the same number of units as the
alphabetical numeral contains tens, hundreds, thousands, &c.
Thus the •pythmen of 700 (\jr in Greek) is 7 ({) ; that of
j? (6000) is q (6), and so on. The method then proceeded
to find the pythmen of a certain name, say 'Aya/ii/jLucov.
Taking the pythmenes of all the letters and adding them,
we have
1 + 3 + 1+4 + 5 + 4 + 5 + 8 + 5 = 36.
Take the pythmenes of 36, namely 3 and 6, and their sum is
9. The pythmen of 'Aya/iep.vcov is therefore 9. Next take
the name "Ektgup', the pythmenes are 5, 2, 3, 8, 1, the sum of
which is 19 ; the pythmenes of 19 are 1, 9 ; the sum of 1 and
9 is 10, the pythmen of which is 1. The pythmen of "Ektoc>p
is therefore 1. 'It is easier', says Hippolytus,. 'to proceed
thus. Finding the pythmenes of the letters, we obtain, in the
case of "EicToop, 19 as their sum. Divide this by 9 and note
the remainder : thus, if I divide 19 by 9, the remainder is 1,
for nine times 2 is 18, and 1 is left, which will accordingly
be the pythmen of the name "Ektcop' Again, take the name
ndrpoKXos. The sum of the pythmenes is
8+1+3+1+7+2+3+7+2 =34:
and 3 + 4 = 7, so that 7 is the pythmien of IldrpoKXo?.
' Those then who calculate by the rule of nine take one-ninth
of the sum of the pythmenes and then determine the sum of
the 'pythmenes in the remainder. Those on the other hand
who follow the " rule of seven " divide by 7. Thus the sum
of the pythmenes in UdrpoKXos was found to be 34. This,
divided by 7, gives 4, and since 7 times 4 is 28, the remainder
is 6. . . .' ' It is necessary to observe that, if the division
gives an integral quotient (without remainder), . . . the
pythmen is the number 9 itself ' (that is, if the rule of nine is
followed). And so on.
Two things emerge from this fragment. (1) The use of the
pythmen was not appearing for the first time when Apollonius
framed his system for expressing and multiplying large
numbers ; it originated much earlier, with the Pythagoreans.
IAMBL1CHUS 117
(2) The method of calculating the pythmerb is like the opera-
tion of ' casting out nines ' in the proof which goes by that
name, where we take the sum of the digits of a number and
divide by 9 to get the remainder. The method of verification
by ' casting out nines ' came to us from the Arabs, who may,
as Avicenna and Maximus Planudes tell us, have got it from
the Indians ; but the above evidence shows that, at all events,
the elements from which it was built up lay ready to hand
in the Pythagorean arithmetic.
IV
THE EARLIEST GREEK GEOMETRY. THALES
The 'Summary* of Proclus.
We shall often, in the course of this history, have occasion
to quote from the so-called ' Summary ' of Proclus, which has
already been cited in the preceding chapter. Occupying a
few pages (65-70) of Proclus's Commentary on Euclid, Book I,
it reviews, in the briefest possible outline, the course of Greek
geometry from the earliest times to Euclid, with special refer-
ence to the evolution of the Elements. At one time it was
often called the ' Eudemian summary ', on the assumption
that it was an extract from the great History of Geometry in
four Books by Eudemus, the pupil of Aristotle. But a perusal
of the summary itself is sufficient to show that it cannot
have been written by Eudemus ; the most that can be said is
that, down to a certain sentence, it was probably based, more
or less directly, upon data appearing in Eudemus's History,
At the sentence in question there is a break in the narrative,
as follows :
' Those who have compiled histories bring the development
of this science up 'to this point. Not much younger than
these is Euclid, who put together the Elements, collecting
many of the theorems of Eudoxus, perfecting many others by
Theaetetus, and bringing to irrefragable demonstration the
propositions which had only been somewhat loosely proved by
his predecessors.'
Since Euclid was later than Eudemus, it is impossible that
Eudemus can have written this ; while the description ' those
who have compiled histories', and who by implication were
a little older than Euclid, suits Eudemus excellently. Yet the
style of the summary after the break does not show any
such change from that of the earlier portion as to suggest
THE « SUMMARY' OF PROCLUS 119
different authorship. The author of the earlier portion fre-
quently refers to the question of the origin of the Elements of
Geometry in a way in which no one would be likely to write
who was not later than Euclid ; and it seems to be the same
hand which, in the second portion, connects the Elements of
Euclid with the work of Eudoxus and Theaetetus. Indeed
the author, whoever he was, seems to have compiled the sum-
mary with one main object in view, namely, to trace the origin
and growth of the Elements of Geometry; consequently he
omits to refer to certain famous discoveries in geometry such
as the solutions of the problem of the duplication of the cube,
doubtless because they did not belong to the Elements. In
two cases he alludes to such discoveries, as it were in paren-
thesis, in order to recall to the mind of the reader a current
association of the name of a particular geometer with a par-
ticular discovery. Thus he mentions Hippocrates of Chios as
a famous geometer for the particular reason that he was the
first to write Elements, and he adds to his name, for the pur-
pose of identification, ' the discoverer of the quadrature of the
lune \ Similarly, when he says of Pythagoras ' (he it was)
who ' (oy 8tj . . .) ' discovered the theory of irrationals [or
" proportions " ] and the construction of the cosmic figures ',
he seems to be alluding, entirely on his own account, to a
popular tradition to that effect. If the summary is the work
of one author, who was it ? Tannery answers that it was
Geminus ; but this seems highly improbable, for the extracts
from Geminus's work which we possess suggest that the
subjects therein discussed were of a different kind ; they seem
rather to have been general questions relating to the philoso-
phy and content of mathematics, and even Tannery admits
that historical details could only have come incidentally into
the work.
Could the author have been Proclus himself ? This again
seems, on the whole, improbable. In favour of the authorship
of Proclus are the facts (1) that the question of the origin of
the Elements is kept prominent and (2) that there is no men-
tion of Democritus, whom Eudemus would not have ignored,
while a follower of Plato such as Proclus might have done
him this injustice, following the example of Plato himself, who
was an opponent of Democritus, never once mentions him, and
120 THE EARLIEST GREEK GEOMETRY. THALES
is said to have wished to burn all his writings. On the other
hand (1) the style of the summary is not such as to point
to Proclus as the author ; (2) if he wrote it, it is hardly
conceivable that he would have passed over in silence the dis-
covery of the analytical method, ' the finest ', as he says else-
where, of the traditional methods in geometry, * which Plato is
said to have communicated to Laodamas'. Nor (3) is it
easy to suppose that Proclus would have spoken in the
detached way that the author does of Euclid whose Elements
was the subject of his whole commentary : ' Not much younger
than these is Euclid, who compiled the Elements . . . '. ' This
man lived in the time of the first Ptolemy . . .'. On the whole,
therefore, it would seem probable that the body of the sum-
mary was taken by Proclus from a compendium made by some
writer later than Eudemus, though the earlier portion was
based, directly or indirectly, upon notices in Eudemus's History.
But the prelude with which the summary is introduced may
well have been written, or at all events expanded, by Proclus
himself, for it is in his manner to bring in ' the inspired
Aristotle' (6 Scll/xovlos 'ApLo-TOTeX-qs;) — as he calls him here and
elsewhere — and the transition to the story of the Egyptian
origin of geometry may also be his :
' Since, then, we have to consider the beginnings of the arts
and sciences with reference to the particular cycle [of the
series postulated by Aristotle] through which the universe is
at present passing, we say that, according to most accounts,
geometry was first discovered in Egypt, having had its origin
in the measurement of areas. For this was a necessity for the
Egyptians owing to the rising of the Nile which effaced the
proper boundaries of everybody's lands.'
The next sentences also may well be due to Proclus :
' And it is in no way surprising that the discovery of this as
well as the other sciences had its beginning in practical needs,
seeing that everything that is in the course of becoming pro-
gresses from the imperfect to the perfect. Thus the transition
from sensation to reasoning and from reasoning to under-
standing is only natural.'
These sentences look like reflections by Proclus, and the
transition to the summary proper follows, in the words :
'Accordingly, just as exact arithmetic began among the
ORIGIN OF GEOMETRY 121
Phoenicians owing to its use in commerce and contracts, so
geometry was discovered in Egypt for the reason aforesaid.'
Tradition as to the origin of geometry.
Many Greek writers besides Proclus give a similar account
of the origin of geometry. Herodotus says that Sesostris
(Ramses II, circa 1300 B.C.) distributed the land among all the
Egyptians in equal rectangular plots, on which he levied an
annual tax ; when therefore the river swept away a portion
of a plot and the owner applied for a corresponding reduction
in the tax, surveyors had to be sent down to certify what the
reduction in the area had been. ' This, in my opinion (SoKeei
fjioi) ', he continues, ' was the origin of geometry, which then
passed into Greece.' 1 The same story, a little amplified, is
repeated by other writers, Heron of Alexandria, 2 Diodorus
Siculus, 3 and Strabo. 4 True, all these statements (even if that
in Proclus was taken directly from Eudemus's History of
Geometry) may all be founded on the passage of Herodotus,
and Herodotus may have stated as his own inference what he
was told in Egypt ; for Diodorus gives it as an Egyptian
tradition that geometry and astronomy were the discoveries
of Egypt, and says that the Egyptian priests claimed Solon,
Pythagoras, Plato, Democritus, Oenopides of Chios, and
Eudoxus as their pupils. But the Egyptian claim to the
discoveries was never disputed by the Greeks. In Plato's
Pkaedrus Socrates is made to say that he had heard that the
Egyptian god Theuth was the first to invent arithmetic, the
science of calculation, geometry, and astronomy. 5 Similarly
Aristotle says that the mathematical arts first took shape in
Egypt, though he. gives as the reason, not the practical need
which arose for a scientific method of measuring land, but the
fact that in Egypt there was a leisured class, the priests, who
could spare time for such things. 6 Democritus boasted that no
one of his time had excelled him ' in making lines into figures
and proving their properties, not even the so-called Harpe-
donaptae in Egypt '. 7 This word, compounded of two Greek
words, dprreSovrj and oltttuv, means ' rope-stretchers ' or ' rope-
1 Herodotus ii. 109. 2 Heron, Geoni. c. 2, p. 176, Heib.
3 Diod. Sic. i. 69, 81. 4 Strabo xvii. c. 3.
5 Plato, Phaedrus 274 c. 6 Arist. Metaph. A. 1, 981 b 28.
7 Clem. Strom, i. 15. 69 (Vorsokratiker, ii 3 , p. 128. 5-7).
122 THE EARLIEST GREEK GEOMETRY. THALES
fasteners'; and, while it is clear from the passage that the
persons referred to were clever geometers, the word reveals a
characteristic modus operandi. The Egyptians were ex-
tremely careful about the orientation of their temples, and
the use of ropes and pegs for marking out the limits,
e.g. corners, of the sacred precincts is portrayed in all
pictures of the laying of foundation stones of temples. 1 The
operation of ' rope-stretching ' is mentioned in an inscription on
leather in the Berlin Museum as having been in use as early
as Amenemhat I (say 2300 B.C.). 2 Now it was the practice
of ancient Indian and probably also of Chinese geometers
to make, for instance, a right angle by stretching a rope
divided into three lengths in the ratio of the sides of a right-
angled triangle in rational numbers, e.g. 3, 4, 5, in such a way
that the three portions formed a triangle, when of course a right
angle would be formed at the point where the two smaller
sides meet. There seems to be no doubt that the Egyptians
knew that the triangle (3, 4, 5), the sides of which are so
related that the square on the greatest side is equal to the
sum of the squares on the other two, is right-angled ; if this
is so, they were acquainted with at least one case of the
famous proposition of Pythagoras.
Egyptian geometry, i. e. mensuration.
We might suppose, from Aristotle's remark about the
Egyptian priests being the first to cultivate mathematics
because they had leisure, that their geometry would have
advanced beyond the purely practical stage to something
more like a theory or science of geometry. But the docu-
ments which have survived do not give any ground for this
supposition ; the art of geometry in the hands of the priests
never seems to have advanced beyond mere routine. The
most important available source of information about Egyptian
mathematics is the Papyrus Rhind, written probably about
1700 B.C. but copied from an original of the time of King
Amenemhat III (Twelfth Dynasty), say 2200 B.C. The geo-
metry in this ' guide for calculation, a means of ascertaining
everything, of elucidating all obscurities, all mysteries, all
1 Brugsch, Steininschrift und Blbehoort y 2nd ed., p. 36.
2 Dumichen, Denderatempel, p. 33.
EGYPTIAN GEOMETRY 123
difficulties', as it calls itself, is rough mensuration. The
following are the cases dealt with which concern us here.
(1) There is the rectangle, the area of which is of course
obtained by multiplying together the numbers representing
the sides. (2) The measure of a triangle is given as the pro-
duct of half the base into the side. And here there is a differ-
ence of opinion as to the kind of triangle measured. Eisenlohr
and Cantor, taking the diagram to represent an isosceles tri-
angle rather inaccurately drawn, have to assume error on
the part of the writer in making the area \ab instead of
^aV(b 2 — f a 2 ) where a is the base and b the 'side ', an error
which of course becomes less serious as a becomes smaller
relatively to b (in the case taken a = 4, b = 10, and the area
as given according to the rule, i.e. 20, is not greatly different
from the true value 19-5959). But other authorities take the
triangle to be right-angled and b to be the side perpendicular
to the base, their argument being that the triangle as drawn
is not a worse representation of a right-angled triangle than
other triangles purporting to be right-angled which are found
in other manuscripts, and indeed is a better representation of
a right-angled triangle than it is of an isosceles triangle, while
the number representing the side is shown in the figure along-
side one only of the sides, namely that adjacent to the angle
which the more nearly represents a right angle. The advan-
tage of this interpretation is that the rule is then correct
instead of being more inaccurate than one would expect from
a people who had expert land surveyors to measure land for
the purpose of assessing it to tax. The same doubt arises
with reference to (3) the formula for the area of a trapezium,
namely ^ (a + c) x b, where a, c are the base and the opposite
parallel side respectively, while b is the ' side ', i.e. one of the
non-parallel sides. In this case the figure seems to have been
intended to be isosceles, whereas the formula is only accurate
if b, one of the non-parallel sides, is at right angles to the base,
in which case of course the side opposite to b is not ; at right
angles to the base. As the parallel sides (6, 4) in the case
taken are short relatively to the ' side ' (20), the angles at the
base are not far short of being right angles, and it is possible
that one of them, adjacent to the particular side which is
marked 20, was intended to be right. The hypothesis that
124 THE EARLIEST GREEK GEOMETRY. THALES
the triangles and trapezia are isosceles, and that the formulae
are therefore crude and inaccurate, was thought to be con-
firmed by the evidence of inscriptions on the Temple of Horus
at Edfu. This temple was planned out in 237 B.C.; the in-
scriptions which refer to the assignment of plots of ground to
the priests belong to the reign of Ptolemy XI, Alexander I
(107-88 B.C.). From so much of these inscriptions as were
published by Lepsius 1 we gather that \ (a + c) . ■§ (6 + d) was a
formula for the area of a quadrilateral the sides of which in
order are a, b, c, d. Some of the quadrilateral figures are
evidently trapezia with the non-parallel sides equal ; others are
not, although they are commonly not far from being rectangles
or isosceles trapezia. Examples are ' 16 to 15 and 4 to 3f make
58J' (i.e. -I (16 + 15) X |(4 + 3§) = 58|); < 9f to 1 Of and 24f f to
22| | make 236£ ' ; '22 to 23 and 4 to 4 make 90 ', and so on.
Triangles are not made the subject of a separate formula, but
are regarded as cases of quadrilaterals in which the length of
one side is zero. Thus the triangle 5, 17, 17 is described as a
figure with sides * to 5 and 17 to 17', the area being accord-
ingly \ (0 + 5) . -|(1 7 + 1 7) or 42J ; is expressed by hieroglyphs
meaning the word Nen. It is remarkable enough that the use
of a formula so inaccurate should have lasted till 200 years or
so after Euclid had lived and taught in Egypt ; there is also
a case of its use in the Liber Geeponicus formerly attributed to
Heron, 2 the quadrilateral having two opposite sides parallel
and the pairs of opposite sides being (32, 30) and (18, 16). But
it is right to add that, in the rest of the Edfu inscriptions
published later by Brugsch, there are cases where the inaccu-
rate formula is not used, and it is suggested that what is being
attempted in these cases is an approximation to the square
root of a non-square number. 3
We come now (4) to the mensuration of circles as found
in the Papyrus Rhind. If d is the diameter, the area is
given as {(1 — i)d} 2 or §y<i 2 . As this is the corresponding
figure to -fffd 2 , it follows that the value of ir is taken as
- 2 g 5 j 6 - = (~§-) 2 , or 3-16, very nearly. A somewhat different
value for w has been inferred from measurements of certain
1 'Ueber eine hieroglyphische Inschrift am Tempel von Edfu' (Abh.
der Berliner Akad., 1855, pp. 69-114).
2 Heron, ed. Hultsch, p. 212. 15-20 (Heron, Geom. c. 6. 2, Heib.).
3 M. Simon, Gesch. d. Math, im Alterium, p. 48.
EGYPTIAN GEOMETRY 125
heaps of grain or of spaces which they fill. Unfortunately
the shape of these spaces or heaps cannot be determined with
certainty. The word in the Papyrus Rhind is shaa ; it is
evident that it ordinarily means a rectangular parallelepiped,
but it can also be applied to a figure with a circular base,
e. g. a cylinder, or a figure resembling a thimble, i. e. with
a rounded top. There is a measurement of a mass of corn
apparently of the latter sort in one of the Kahun papyri. 1
The figure shows a circle with 1365§ as the content of the
heap written within it, and with 12 and 8 written above and
to the left of the circle respectively. The calculation is done
in this way. 1 2 is taken and § of it added ; this gives 1 6 ;
16 is squared, which gives 256, and finally 256 is multiplied
by •§ of 8, which gives 13 65 J. If for the original figures
12 and 8 we write h and k respectively, the formula used for
the content is (§ A) 2 . ■§/<;. Griffith took 12 to be the height
of the figure and 8 to be the diameter of the base. But
according to another interpretation, 2 12 is simply § of 8, and
the figure to be measured is a hemisphere with diameter
8 ells. If this is so, the formula makes the content of a
hemisphere of diameter k to be (f . §&) 2 .§& or §& 3 . Com-
paring this with the true volume of the hemisphere, f . §7r/c 3
or ^irh 3 = 134*041 cubic ells, we see that the result 1365 J
obtained by the formula must be expressed in xo^ ns of a cubic
ell : consequently for ^it the formula substitutes 3%, so that
the formula gives 3-2 in place of it, a value different from the
3-16 of Ahmes. Borchardt suggests that the formula for the
measurement of a hemisphere was got by repeated practical
measurements of heaps of corn built up as nearly as possible
in that form, in which case the inaccuracy in the figure for it
is not surprising. With this problem from the Kahun papyri
must be compared No. 43 from the Papyrus Rhind. A curious
feature in the measurements of stores or heaps of corn in
the Papyrus Rhind is the fact, not as yet satisfactorily ex-
plained, that the area of the base (square or circular) is first
found and is then regularly multiplied, not into the ' height '
itself, but into •§ times the height. But in No. 43 the calcula-
tion is different and more parallel to the case in the Kahun
papyrus. The problem is to find the content of a space round
1 Griffith, Kahun Papyri, Pt. I, Plate 8. 2 Simon, 7. c.
126 THE EARLIEST GREEK GEOMETRY. THALES
in form ' 9 in height and 6 in breadth '. The word qa, here
translated ' height ', is apparently used in other documents
for ' length ' or ' greatest dimension ', and must in this case
mean the diameter of the base, while the ' breadth ' is the
height in our sense. If we denote the diameter of the circular
base by k, and the height by h, the formula used in this
problem for finding the volume is (§• §&) 2 . ■§&. Here it is
not § h, but %h, which is taken as the last factor of the
product. Eisenlohr suggests that the analogy of the formula
for a hemisphere, 7rr 2 .§r, may have operated to make the
calculator take § of the height, although the height is not
in the particular case the same as the radius of the base, but
different. But there remains the difficulty that (f) 2 or -^
times the area of the circle of diameter k is taken instead
of the area itself. As to this Eisenlohr can only suggest that
the circle of diameter k which was accessible for measurement
was not the real or mean circular section, and that allowance
had to be made for this, or that the base was not a circle of
diameter k but an ellipse with -^ k and k as major and minor
axes. But such explanations can hardly be applied to the
factor (§) 2 in the Kahun case if the latter is really the case
of a hemispherical space as suggested. Whatever the true
explanation may be, it is clear that these rules of measure-
ment must have been empirical and that there was little or
no geometry about them.
Much more important geometrically are certain calculations
with reference to the proportions of pyramids (Nos. 56-9 of
the Papyrus Rhind) and a monu-
^fly ment (No. 60). In the case
\ of the pyramid two lines in the
.j\c figure are distinguished, (1)
/\\ ukha-thebt, which is evidently
/ V some line in the base, and
\ I , i - s V (2) piv-em-us or per-em-us
\ I /' " ^\ ('height'), a word from which
Y — ^_ the name Trvpa\xis may have
been derived. 1 The object of
1 Another view is that the words Trypans and nvpapovs, meaning a kind
of cake made from roasted wheat and honey, are derived from nvpoi,
1 wheat ', and are thus of purely Greek origin.
MEASUREMENT OF PYRAMIDS 127
the problems is to find a certain relation called se-qet,
literally ' that which makes the nature ', i. e. that which
determines the proportions of the pyramid. The relation
hukha-thebt T .. „ ., , ,
se-qet = - — : . In the case ot the monument we have
piremus
two other names for lines in the figure, (1) senti, ' foundation ',
or base, (2) qay en heru, 'vertical length', or height; the
same term se-qet is used for the relation — - 7 or
1 • qay en heru
the same inverted. Eisenlohr and Cantor took the lines
(1) and (2) in the case of the pyramid to be different from
the lines (1) and (2) called by different names in the monument.
Suppose A BCD to be the square base of a pyramid, E its
centre, H the vertex, and F the middle point of the side AD
of the base. According to Eisenlohr and Cantor the uleha-
thebt is the diagonal, say AC, of the base, and the pir-em-us
is the edge, as AH. On this assumption the se-qet
AE T7AV
= a tt= cos BAE.
AH
In the case of the monument they took the senti to be the
side of the base, as AB, the qay en heru to be the height of
the pyramid EH, and the se-qet to be the ratio of EH to
^AB or of EH to EF, i.e. the tangent of the angle HFE
which is the slope of the faces of the pyramid. According
to Eisenlohr and Cantor, therefore, the one term se-qet was
used in two different senses, namely, in Nos. 56—9 for cos HAE
and in No. 60 for tan HFE. Borchardt has, however, proved
that the se-qet in all the cases has one meaning, and represents
the cotangent of the slope of the faces of the pyramid,
i. e. cot HFE or the ratio of FE to EH. There is no difficulty
in the use of the different words ukha-thebt and senti to
express the same thing, namely, the side of the base, and
of the different words per-em-us and qay en heru in the same
sense of ' height ' ; such synonyms are common in Egypt, and,
moreover, the word wer used of the pyramids is different
from the word an for the monument. Again, it is clear that,
while the slope, the angle HFE, is what the builder would
want to know, the cosine of the angle HAE, formed by the
edge with the plane of the base, would be of no direct use
128 THE EARLIEST GREEK GEOMETRY. THALES
to him. But, lastly, the se-qet in No. 56 is ff and, if te-qet
is taken in the sense of cot HFE i this gives for the angle
HFE the value of 54° 14/ 16", which is precisely, to the
seconds, the slope of the lower half of the southern stone
pyramid of Dakshur; in Nos. 57-9 the se-qet, J, is the co-
tangent of an angle of 53° 7 / 48", which again is exactly the
slope of the second pyramid of Gizeh as measured by Flinders
Petrie ; and the se-qet in No. 60, which is £, is the cotangent
of an angle of 75° 57' 50", corresponding exactly to the slope
of the Mastaba-tombs of the Ancient Empire and of the
sides of the Medum pyramid. 1
These measurements of se-qet indicate at all events a rule-
of -thumb use of geometrical proportion, and connect themselves
naturally enough with the story of Thales's method of measuring
the heights of pyramids.
The beginnings of Greek geometry.
At the beginning of the summary of Proclus we are told
that THALES (624-547 B.C.)
'first went to Egypt and thence introduced this study
(geometry) into Greece. He discovered many propositions
himself, and instructed his successors in the principles under-
lying many others, his method of attack being in some cases
more general (i. e. more theoretical or scientific), in others
more empirical (alo-OrjTiKdorepoi', more in the nature of simple
inspection or observation).' 2
With Thales, therefore, geometry first becomes a deductive
science depending on general propositions ; this agrees with
what Plutarch says of him as one of the Seven Wise Men :
' he was apparently the only one of these whose wisdom
stepped, in speculation, beyond the limits of practical utility :
the rest acquired the reputation of wisdom in politics.' 3
(Not that Thales was inferior to the others in political
wisdom. Two stories illustrate the contrary. He tried to
save Ionia by urging the separate states to form a federation
1 Flinders Petrie, Pyramids and Temples of Gizeh, p. 162.
2 Proclus on Eucl. I, p. 65. 7-11.
3 Plutarch, Solon, c. 3.
MEASUREMENT OF PYRAMIDS 129
with a capital at Teos, that being the most central place in
Ionia. And when Croesus sent envoys to Miletus to propose
an alliance, Thales dissuaded his fellow-citizens from accepting
the proposal, with the result that, when Cyrus conquered, the
city was saved.)
(a) Measurement of height of pyramid.
The accounts of Thales's method of measuring the heights
of pyramids vary. The earliest and simplest version is that
of Hieronymus, a pupil of Aristotle, quoted by Diogenes
Laertius :
' Hieronymus says that he even succeeded in measuring the
pyramids by observation of the length of their shadow at
the moment when our shadows are equal to our own height.' *
Pliny says that
' Thales discovered how to obtain the height of pyramids
and all other similar objects, namely, by measuring the
shadow of the object at the time when a body and its shadow
are equal in length.' 2
Plutarch embellishes the story by making Niloxenus say
to Thales :
' Among other feats of yours, he (Amasis) was particularly
pleased with your measurement of the pyramid, when, without
trouble or the assistance of any instrument, you merely set
up a stick at the extremity of the shadow cast by the
pyramid and, having thus made two triangles by the impact
of the sun's rays, you showed that the pyramid has to the
stick the same ratio which the shadow has to the shadow.' 3
The first of these versions is evidently the original one and,
as the procedure assumed in it is more elementary than the
more general method indicated by Plutarch, the first version
seems to be the more probable. Thales could not have failed
to observe that, at the time when the shadow of a particular
object is equal to its height, the same relation holds for all
other objects casting a shadow ; this he would probably
infer by induction, after making actual measurements in a
1 Diog. L. i. 27. 2 N. H. xxxvi. 12 (17).
3 Plut. Conv. sept. sap. 2, p. 147 a.
1523 K
130 THE EARLIEST GREEK GEOMETRY. THALES
considerable number of cases at a time when he found the
length of the shadow of one object to be equal to its height.
But, even if Thales used the more general method indicated
by Plutarch, that method does not, any more than the Egyptian
se-qet calculations, imply any general theory of similar tri-
angles or proportions ; the solution is itself a se-qet calculation,
just like that in No. 57 of Ahmes's handbook. In the latter
problem the base and the se-qet fire given, and we have to
find the height. So in Thales's problem we get a certain
se-qet by dividing the measured length of the shadow of the
stick by the length of the stick itself ; we then only require
to know the distance between the point of the shadow corre-
sponding to the apex of the pyramid and the centre of the
base of the pyramid in order to determine the height; the
only difficulty would be to measure or estimate the distance
from the apex of the shadow to the centre of the base.
(f3) Geometrical theorems attributed to Thales.
The following are the general theorems in elementary
geometry attributed to Thales.
(1) He is said to have been the first to demonstrate that
a circle is bisected by its diameter. 1
(2) Tradition credited him with the first statement of the
theorem (Eucl. I. 5) that the angles at the base of any
isosceles triangle are equal, although he used the more archaic
term ' similar ' instead of ' equal '. 2
(3) The proposition (Eucl. I. 15) that, if two straight lines
cut one another, the vertical and opposite angles are equal
was discovered, though not scientifically proved, by Thales.
Eudemus is quoted as the authority for this. 3
(4) Eudemus in his History of Geometry referred to Thales
the theorem of Eucl. I. 26 that, if two triangles have two
angles and one side respectively equal, the triangles are equal
in all respects.
' For he (Eudemus) says that the method by which Thales
showed how to find the distances of ships from the shore
necessarily involves the use of this theorem.' 4
1 Proclus on Eucl. I, p. 157. 10. 2 16., pp. 250. 20-251. 2.
3 lb., p. 299. 1-5. 4 lb., p. 352. 14-18.
GEOMETRICAL THEOREMS
131
(5) ' Pamphile says that Thales, who learnt geometry from
the Egyptians, was the first to describe on a circle a triangle
(which shall be) right-angled (KaTaypdyjrat kvkXov to Tpiyoavov
opOoycouiou), and that he sacrificed an ox (on the strength of
the discovery). Others, however, including Apollodorus the
calculator, say that it was Pythagoras.' 1
The natural interpretation of Pamphile's words is to suppose
that she attributed to Thales the discovery that the angle
in a semicircle is a right angle.
Taking these propositions in order, we may observe that,
when Thales is said to have ' demonstrated ' (ccTroSeT^aL) that
a circle is bisected by its diameter, whereas he only ' stated '
the theorem about the isosceles triangle and ' discovered ',
without scientifically proving, the equality of vertically
opposite angles, the word ' demonstrated ' must not be taken
too literally. Even Euclid did not 'demonstrate' that.a circle
is bisected by its diameter, but merely stated the fact in
I. Def. 17. Thales therefore probably
observed rather than proved the property ;
and it may, as Cantor says, have been
suggested by the appearance of certain
figures of circles divided into a number
of equal sectors by 2, 4, or 6 diameters
such as are found on Egyptian monu-
ments or represented on vessels brought
by Asiatic tributary kings in the time of the eighteenth
dynasty. 2
It has been suggested that the use of the word ' similar ' to
describe the equal angles of an isosceles triangle indicates that
Thales did not yet conceive of an angle as a magnitude, but
as a figure having a certain sha^e, a view which would agree
closely with the idea of the Egyptian se-qet, ' that which
makes the nature ', in the sense of determining a similar or
the same inclination in the faces of pyramids.
With regard to (4), the theorem of Eucl. I. 26, it will be
observed that Eudemus only inferred that this theorem was
known to Thales from the fact that it is necessary to Thales's
determination of the distance of a ship from the shore.
Unfortunately the method used can only be conjectured.
1 Diog. L. i. 24, 25.
2 Cantor, Gesch. d. Math. i s , pp. 109, 140.
K 2
132 THE EARLIEST GREEK GEOMETRY. THALES
The most usual supposition is that Thales, observing the ship
from the top of a tower on the sea-shore, used the practical
equivalent of the proportionality of the sides of two similar
right-angled triangles, one small and one large. Suppose B
to be the base of the tower, C the ship. It was only necessary
for a man standing at the top of the
tower to have an instrument with
two legs forming a right angle, to
place it with one leg DA vertical and
in a straight line with B, and the
other leg DE in the direction of the
ship, to take any point A on DA,
and then to mark on DE the point E
where the line of sight from A to C cuts the leg DE. Then
AD (= I, say) and DE (= m, say) can be actually measured,
as also the height BD (= h, say) from D to the foot of the
tower, and, by similar triangles, t
BC=(h + l).j.
The objection to this solution is that it does not depend
directly on Eucl. I. 26, as Eudemus implies. Tannery 1 there-
fore favours the hypothesis of a solution on the lines followed
by the Roman agrimensor Marcus Junius Nipsus in his
fluminis varatio. — To find the distance from
A to an inaccessible point B. Measure from A ,
along a straight line at right angles to AB,
a distance AC, and bisect it at D. From C, on
the side of AC remote from B, draw CE at
right angles to AC, and let E be the point on
it which is in a straight line with B and D.
Then clearly, by Eucl. I. 26, CE is equal to
AB; and CE can be measured, so that AB
is known.
This hypothesis is open to a different objec-
tion, namely that, as a rule, it would be
difficult, in the supposed case, to get a sufficient amount of
free and level space for the construction and measurements.
I have elsewhere 2 suggested a still simpler method free
1 Tannery, La geometrie grecque, pp. 90-1.
2 The Thirteen Books of Euclid's Elements , vol. i, p. 305.
DISTANCE OF A SHIP AT SEA 133
from this objection, and depending equally directly on Eucl.
I. 26. If the observer was placed on the top of a tower, he
had only to use a rough instrument made of a straight stick
and a cross-piece fastened to it so as to be capable of turning
about the fastening (say a nail) so that it could form any
angle with the stick and would remain where it was put.
Then the natural thing would be to fix the stick upright (by
means of a plumb-line) and direct the cross-piece towards the
ship. Next, leaving the cross-piece at the angle so found,
he would turn the stick round, while keeping it vertical, until
the cross-piece pointed to some visible object on the shore,
which would be mentally noted ; after this it would only
be necessary to measure the distance of the object from the
foot of the tower, which distance would, by Eucl. I. 26, be
equal to the distance of the ship. It appears that this precise
method is found in so many practical geometries of the first
century of printing that it must be assumed to have long
been a common expedient. There is a story that one of
Napoleon's engineers won the Imperial favour by quickly
measuring, in precisely this way, the width of a stream that
blocked the progress of the army. 1
There is even more difficulty about the dictum of Pamphile
implying that Thales first discovered the fact that the angle
in a semicircle is a right angle. Pamphile lived in the reign
of Nero (a. d. 54-68), and is therefore a late authority. The
date of Apollodorus the 'calculator' or arithmetician is not
known, but he is given as only one of several authorities who
attributed the proposition to Pythagoras. Again, the story
of the sacrifice of an ox by Thales on the occasion of his
discovery is suspiciously like that told in the distich of
Apollodorus ' when Pythagoras discovered that famous pro-
position, on the strength of which he offered a splendid
sacrifice of oxen '. But, in quoting the distich of Apollodorus,
Plutarch expresses doubt whether the discovery so celebrated
was that of the theorem of the square of the hypotenuse or
the solution of the problem of ' application of areas ' 2 ; there
is nothing about the discovery of the fact of the angle in
a semicircle being a right angle. It may therefore be that
1 David Eugene Smith, The Teaching of Geometry, pp. 172-3.
2 Plutarch, Non posse suaviter rivi secundum Epicurum, c. 11, p. 1094 b.
134 THE EARLIEST GREEK GEOMETRY. THALES
Diogenes Laertius was mistaken in bringing Apollodorus into
the story now in question at all ; the mere mention of the
sacrifice in Pamphile's account would naturally recall Apollo-
dorus's lines about Pythagoras, and Diogenes may have
forgotten that they referred to a different proposition.
But, even if the story of Pamphile is accepted, there are
difficulties of substance. As Allman pointed out, if Thales
knew that the angle in a semicircle
is a right angle, he was in a position
at once to infer that the sum of the
angles of any right-angled triangle is
equal to two right angles. For suppose
that BC is the diameter of the semi-
circle, the centre, and A a point on
the semicircle ; we are then supposed
to know that the angle B A C is a right angle. Joining OA,
we form two isosceles triangles OAB, OAC; and Thales
knows that the base angles in each of these triangles are
equal. Consequently the sum of the angles OAB, OAC is
equal to the sum of the angles OB A, OCA. The former sum
is known to be a right angle ; therefore the second sum is
also a right angle, and the three angles of the triangle ABC
are together equal to twice the said sum, i. e. to two right
angles.
Next it would easily be seen that any triangle can be
divided into two right-angled triangles by drawing a perpen-
dicular AD from a vertex A to the
opposite side BC. Then the three
angles of each of the right-angled
triangles ABD, ADC are together equal
to two right angles. By adding together
the three angles of both triangles we
find that the sum of the three angles of the triangle ABC
together with the angles ADB, ADC is equal to four right
angles; and, the sum of the latter two angles being two
right angles, it follows that the sum of the remaining angles,
the angles at A, B, C, is equal to two right angles. And ABC
is any triangle.
Now Euclid in III. 31 proves that the angle in a semicircle
is a right angle by means of the general theorem of I. 32
THE ANGLE IN A SEMICIRCLE
135
that the sum of the angles of any triangle is equal to two
right angles ; but if Thales was aware of the truth of the
latter general proposition and proved the proposition about
the semicircle in this way, by means of it, how did Eudemus
come to credit the Pythagoreans, not only with the general
proof, but with the discovery, of the theorem that the angles
of any triangle are together equal to two right angles 1 1
Cantor, who supposes that Thales proved his proposition
after the'manner of Euclid III. 31, i.e. by means of the general
theorem of I. 32, suggests that Thales arrived at the truth of
the latter, not by a general proof like that attributed by
Eudemus to the Pythagoreans, but by an argument following
the steps indicated by Geminus. Geminus says that
' the ancients investigated the theorem of the two right
angles in each individual species of triangle, first in the equi-
lateral, then in the isosceles, and afterwards in the scalene
triangle, but later geometers demonstrated the general theorem
that in any triangle the three interior angles are equal to two
right angles '. 2
The ' later geometers ' being the Pythagoreans, it is assumed
that the 'ancients' may be Thales and his contemporaries.
As regards the equilateral triangle, the fact might be suggested
by the observation that six such triangles arranged round one
point as common vertex would fill up the space round that
point ; whence it follows that each angle is one-sixth of four
right angles, and three such angles make up two right angles.
Again, suppose that in either an equilateral or an isosceles
triangle the vertical angle is bisected by a straight line meet-
ing the base, and that the rectangle of which the bisector and
one half of the base are adjacent sides is completed ; the
rectangle is double of the half of the original triangle, and the
angles of the half-triangle are together equal to half the sum
1 Proclus on Eucl. I, p. 379. 2-5.
2 See Eutocius, Coinm. on Conies of Apollonius (vol. ii, p. 170, Heib.).
136 THE EARLIEST GREEK GEOMETRY. THALES
of the angles of the rectangle, i.e. are equal to two right
angles ; and it immediately follows that the sum of the angles
of the original equilateral or isosceles triangle is equal to two
right angles. The same thing is easily proved of any triangle
by dividing it into two right-angled
triangles and completing the rectangles
which are their doubles respectively, as
in the figure. But the fact that a proof
on these lines is just as easy in the case
of the general triangle as it is for the
equilateral and isosceles triangles throws doubt on the whole
procedure ; and we are led to question whether there is any
foundation for Geminus's account at all. Aristotle has a re-
mark that
' even if one should prove, with reference to each (sort of)
triangle, the equilateral, scalene, and isosceles, separately, that
each has its angles equal to two right angles, either by one
proof or by different proofs, he does not yet know that the
triangle, i.e. the triangle in general, has its angles equal to
two right angles, except in a sophistical sense, even though
there exists no triangle other than triangles of the kinds
mentioned. For he knows it not qua triangle, nor of every
triangle, except in a numerical sense ; he does not know it
notionally of every triangle, even though there be actually no
triangle which he does not know \*
It may well be that Geminus was misled into taking for
a historical fact what Aristotle gives only as a hypothetical
illustration, and that the exact stages by which the proposi-
tion was first proved were not those indicated by Geminus.
Could Thales have arrived at his proposition about the
semicircle without assuming, or even knowing, that the sum
of the angles of any triangle is equal to two right angles 1 It
seems possible, and in the following way.
Many propositions were doubtless first
discovered by drawing all sorts of figures
and lines inthem,andobservinga|9jpare / ni
relations of equality, &c, between parts.
It would, for example, be very natural
to draw a rectangle, a figure with four right! angles (which, it
1 Arist. Anal. Post. i. 5, 74 a 25 sq.
THE ANGLE IN A SEMICIRCLE 137
would be found, could be drawn in practice), and to put in the
two diagonals. The equality of the opposite sides would
doubtless, in the first beginnings of geometry, be assumed as
obvious, or verified by measurement. If then it was assumed
that a rectangle is a figure with all its angles right angles and
each side equal to its opposite, it would be easy to deduce
certain consequences. Take first the two triangles ADC, BCD.
Since by hypothesis AD = BC and CD is common, the two
triangles have the sides AD, DC respectively equal to the sides
BC, CD, and the included angles, being right angles, are equal ;
therefore the triangles ADC, BCD are equal in all respects
(cf. Eucl. I. 4), and accordingly the angles ACD (i.e. OCD) and
BDC (i.e. ODC) are equal, whence (by the converse of Eucl. I. 5,
known to Thales) OD = OC. Similarly by means of the
equality of AB, CD we prove the equality of OB, OC. Conse-
quently OB, OC, OD (and OA) are all equal. It follows that
a circle with centre and radius OA passes through B, C, D
also ; since AO, OC are in a straight line, AC is a diameter of
the circle, and the angle ABC, by hypothesis a right angle, is
an ' angle in a semicircle '. It would then appear that, given
any right angle as ABC standing on iC as base, it was only
necessary to bisect AC at 0, and would then be the centre of
a semicircle on iC as diameter and passing through B. The
construction indicated would be the construction of a circle
about the right-angled triangle ABC, which seems to corre-
spond well enough to Pamphile's phrase about ' describing on
(i. e. in) a circle a triangle (which shall be) right angled '.
(y) Thales as astronomer.
Thales was also the first Greek astronomer. Every one
knows the story of his falling into a well when star-gazing,
and being rallied by 'a clever and pretty maidservant from
Thrace ' for being so eager to know what goes on in the
heavens that he could not see what was straight in front
of him, nay, at his very feet. But he was not merely a star-
gazer. There is good evidence that he predicted a solar eclipse
which took place on May 28, 585 B.C. We can conjecture
the basis of this prediction. The Babylonians, as the result
of observations continued through centuries, had discovered
the period of 223 lunations after which eclipses recur; and
138 THE EARLIEST GREEK GEOMETRY. THALES
this period was doubtless known to Thales, either directly or
through the Egyptians as intermediaries. Thales, however,
cannot have known the cause of eclipses ; he could not have
given the true explanation of lunar eclipses (as the Duxo-
graphi say he did) because he held that the earth is a circular
disc floating on the water like a log ; and, if he had correctly
accounted for solar eclipses, it is impossible that all the
succeeding Ionian philosophers should, one after another, have
put forward the fanciful explanations which we find recorded.
Thales's other achievements in astronomy can be very
shortly stated. Eudemus attributed to him the discovery of
' the fact that the period of the sun with reference to the
solstices is not always the same' 1 ; the vague phrase seems
to mean that he discovered the inequality of the length of
the four astronomical seasons, that is, the four parts of the
'tropical' year as divided by the solstices and equinoxes.
Eudemus presumably referred to the written works by Thales
On the Solstice and On the Equinoxes mentioned by Diogenes
Laertius. 2 He knew of the division of the year into 365 days,
which he probably learnt from Egypt.
Thales observed of the Hyades that there were two of
them, one north and the other south. He used the Little
Bear as a means of finding the pole, and advised the Greeks
to sail by the Little Bear, as the Phoenicians did, in preference
to their own practice of sailing by the Great Bear. This
instruction was probably noted in the handbook under the
title of Nautical Astronomy, attributed by some to Thales
and by others to Phocus of Samos.
It became the habit of the Doxographi to assign to Thales,
in common with other astronomers in each case, a number
of discoveries not made till later. The following is the list,
with the names of the astronomers to whom the respective
discoveries may with most certainty be attributed : ( 1 ) the
fact that the moon takes its light from the sun (Anaxagoras
and possibly Parmenides) ; (2) the sphericity of the earth
(Pythagoras) ; (3) the division of the heavenly sphere into
five zones (Pythagoras and Parmenides) ; (4) the obliquity
of the ecliptic (Oenopides of Chios); (5) the estimate of the
1 See Theon of Smyrna, p. 198. 17. 2 Diog. L. i. 23.
THALES AS ASTRONOMER 139
sun's diameter as 1/7 2 Oth part of the sun's circle (Aristarchus
of Samos).
From Thales to Pythagoras.
We are completely in the dark as to the progress of geometry
between the times of Thales and Pythagoras. Anaximander
(born about 611/10 B.C.) put forward some daring and original
hypotheses in astronomy. According to him the earth is
a short cylinder with two bases (on one of which we live) and
of depth equal to one-third of the diameter of either base.
It is suspended freely in the middle of the universe without
support, being kept there in equilibrium by virtue of its
equidistance from the extremities and from the other heavenly
bodies all round. The sun, moon, and stars are enclosed in
opaque rings of compressed air concentric with the earth and
filled with fire ; what we see is the fire shining through vents
(like gas-jets, as it were). The sun's ring is 27 or 28 times, the
moon's ring 19 times, as large as the earth, i.e. the sun's
and moon's distances are estimated in terms (as we may
suppose) of the radius of the circular face of the earth ; the
fixed stars and the planets are nearer to the earth than
the sun and moon. This is the first speculation on record
about sizes and distances. Anaximander is also said to have
introduced the gnomon (or sun-dial with a vertical needle)
into Greece and to have shown on it the solstices, the times,
the seasons, and the equinox 1 (according to Herodotus 2 the
Greeks learnt the use of the gnomon from the Babylonians).
He is also credited, like Thales before him, with having
constructed a sphere to represent the heavens. 3 But Anaxi-
mander has yet another claim to undying fame. He was the
first who ventured to draw a map of the inhabited earth.
The Egyptians had drawn maps before, but only of particular
districts ; Anaximander boldly planned out the whole world
with ' the circumference of the earth and sea '. 4 This work
involved of course an attempt to estimate the dimensions of
the earth, though we have no information as to his results.
It is clear, therefore, that Anaximander was something of
1 Euseb. Praep. Evang. x. 14. 11 {Vors. i 3 , p. 14. 28).
2 Hdt. ii. 109. a Diog. L. ii. 2.
4 Diog. L. I. c.
140 THE EARLIEST GREEK GEOMETRY. THALES
a mathematician ; but whether he contributed anything to
geometry as such is uncertain. True, Suidas says that he
' introduced the gnomon and generally set forth a sketch
or outline of geometry ' (oAcoy yecofxerpta? xmorvTrcocriv tSeigev) ;
but it may be that ' geometry ' is here used in its literal sense
of earth-measurement, and that the reference is only to the
famous map.
' Next to Thales, Ameristus, a brother of the poet Stesichorus,
is mentioned as having engaged in the study of geometry ;
and from what Hippias of Elis says it appears that he acquired
a reputation for geometry.' l
Stesichorus the poet lived about 630-550 B.C. The brother
therefore would probably be nearly contemporary with Thales.
We know nothing of him except from the passage of Proelus,
and even his name is uncertain. In Friedlein's edition of
Proelus it is given as Mamercus, after a later hand in cod.
Monac. 427 ; Suidas has it as Mamertinus (s.v. Stesichorus) ;
Heiberg in his edition of Heron's Definitions writes Mamertius,
noting MapfxeTios as the reading of Cod. Paris. Gr. 2385.
1 Proelus on Eucl. I, p. 65. 11-15.
V
PYTHAGOREAN GEOMETRY
The special service rendered by Pythagoras to geometry is
thus described in the Proclus summary :
'After these (Thales and Ameristus or Mamercus) Pythagoras
transformed the study of geometry into a liberal education,
examining the principles of the science from the beginning
and probing the theorems in an immaterial and intellectual
manner : he it was who discovered the theory of irrationals '
(or * proportions ') ' and the construction of the cosmic figures \*
These supposed discoveries will claim our attention pre-
sently ; the rest of the description agrees with another
passage about the Pythagoreans :
' Herein ', says Proclus, ' I emulate the Pythagoreans who
even had a conventional phrase to express what I mean,
" a figure and a platform, not a figure and sixpence ", by
which they implied that the geometry which is deserving of
study is that which, at each new theorem, sets up a platform to
ascend by, and lifts the soul on high instead of allowing it
to go down among sensible objects and so become subser-
vient to the common needs of this mortal life \ 2
In like manner we are told that ' Pythagoras used defini-
tions on account of the mathematical nature of the subject ', 3
which again implies that he took the first steps towards the
systematization of geometry as a subject in itself.
A comparatively early authority, Callimachus (about 250 B.C.),
is quoted by Diodorus as having said that Pythagoras dis-
covered some geometrical problems himself and was the first
to introduce others from Egypt into Greece. 4 Diodorus gives
what appear to be five verses of Callimachus minus a few words ;
1 Proclus on Eucl. I, p. 65. 15-21. 2 lb., p. 84. 15-22.
3 Favorinus in Diog. L. viii. 25.
4 Diodorus x. 6. 4 (Vors. i\ p. 346. 23).
142 PYTHAGOREAN GEOMETRY
a longer fragment including the same passage is now available
(though the text is still deficient) in the Oxyrhynchus Papyri. 1
The story is that one Bathycles, an Arcadian, bequeathed a
cup to be given to the best of the Seven Wise Men. The cup
first went to Thales, and then, after going the round of the
others, was given to him a second time. We are told that
Bathycles's son brought the cup to Thales, and that (presum-
ably on the occasion of the first presentation)
' by a happy chance he found . . . the old man scraping the
ground and drawing the figure discovered by the Phrygian
Euphorbus (= Pythagoras), who was the first of men to draw
even scalene triangles and a circle . . . , and who prescribed
abstinence from animal food '.
Notwithstanding the anachronism, the ' figure discovered by
Euphorbus' is presumably the famous proposition about the
squares on the sides of a right-angled triangle. In Diodorus's
quotation the words after ' scalene triangles ' are kvkXov kirra-
tLr\Kr}{kiTTa\ir]Kz Hunt), which seems unintelligible unless the
' seven-lengthed circle ' can be taken as meaning the ' lengths of
seven circles' (in the sense of the seven independent orbits
of the sun, moon, and planets) or the circle (the zodiac) com-
prehending them all. 2
But it is time to pass on to the propositions in geometry
which are definitely attributed to the Pythagoreans.
1 Oxyrhynchus Papyri, Pt. vii, p. 33 (Hunt).
2 The papyrus has an accent over the e and to the right of the
accent, above the uncertain w, the appearance of a X in dark ink,
x.
thus KaiKVKXoveir, a reading which is not yet satisfactorily explained.
Diels (Vorsokraiiker, i 3 , p. 7) considers that the accent over the « is fatal
to the reading fTrra^Kq, and conjectures kg\ kvkXov e'A(i/ca) KrjSiftagf
vr)(TT€vfiv instead of Hunt's /ml kvkXov eTrfra/^/ce' , rj8e vrjarfvfiv] and
Diodorus's ko\ kvkXov (7rTnv.f]Kn ftifta^ vqo-Tfvav. But kvkXov (Xikci, ' twisted
(or curved) circle ', is very indefinite. It may have been suggested to
Diels by Hermesianax's lines (Athenaeus xiii. 599 A) attributing to
Pythagoras the 'refinements of the geometry of spirals 1 ((Xlkwv ko^o.
yeaffxerpins). One naturally thinks of Plato's dictum (Timaeus 39 A, b)
about the circles of the sun, moon, and planets being twisted into spirals
by the combination of their own motion with that of the daily rotation ;
but this can hardly be the meaning here. A more satisfactory sense
would be secured if we could imagine the circle to be the circle described
about the 'scalene 1 (right-angled) triangle, i.e. if we could take the
reference to be to the discovery of the fact that the angle in a semi-
circle is a right angle, a discovery which, as we have seen, was alterna-
tively ascribed to Thales and Pythagoras.
PYTHAGOREAN GEOMETRY 143
Discoveries attributed to the Pythagoreans.
(a) Equality of the sum of the three angles of a triangle
to two right angles.
We have seen that Thales, if he really discovered that the
angle in a semicircle is a right angle, was in a position, first,
to show that in any right-angled triangle the sum of the three
angles is equal to two right angles, and then, b}^ drawing the
perpendicular from a vertex of any triangle to the opposite
side and so dividing the triangle into two right-angled
triangles, to prove that the sum of the three angles of any
triangle whatever is equal to two right angles. If this method
of passing from the particular case of a right-angled triangle to
that of any triangle did not occur to Thales, it is at any rate
hardly likely to have escaped Pythagoras. But all that we know
for certain is that Eudemus referred to the Pythagoreans
the discovery of the general theorem that in any triangle
the sum of the interior angles is equal to two right angles. 1
Eudemus goes on to tell us how they proved it. The method
differs slightly from that of Euclid, but depends, equally with
Euclid's proof, on the properties of parallels ; it can therefore
only have been evolved at a time when those properties were
already known.
Let ABG be any triangle ; through A draw BE parallel
to BG.
Then, since BG, BE are parallel, the
alternate angles DAB, ABG are equal.
Similarly the alternate angles EAG,
ACB are equal.
Therefore the sum of the angles ABG, Q c
AGB is equal to the sum of the angles BAB, EAG
Add to each sum the angle BAG-, therefore the sum of the
three angles ABG, ACB, BAG, i.e. the three angles of the
triangle, is equal to the sum of the angles DAB, BAG, GAE,
i.e. to two right angles.
We need not hesitate to credit the Pythagoreans with the
more general propositions about the angles of any polygon,
1 Proclus on Eucl. I, p. 397. 2.
144 PYTHAGOREAN GEOMETRY
namely (1) that, if n be the number of the sides or angles, the
interior angles of the polygon are together equal to 2^—4
right angles, and (2) that the exterior angles of the polygon
(being the supplements of the interior angles respectively)
are together equal to four right angles. The propositions are
interdependent, and Aristotle twice quotes the latter. 1 The
Pythagoreans also discovered that the only three regular
polygons the angles of which, if placed together round a com-
mon point as vertex, just fill up the space (four right angles)
round the point are the equilateral triangle, the square, and
the regular hexagon.
(/?) The 'Theorem of Pythagoras' ( = Eucl. I. 47).
Though this is the proposition universally associated by
tradition with the name of Pythagoras, no really trustworthy
evidence exists that it was actually discovered by him. The
comparatively late writers who attribute it to him add the
story that he sacrificed an ox to celebrate his discovery.
Plutarch 2 (born about a.d. 46), Athenaeus 3 (about a.d. 200),
and Diogenes Laertius 4 (a.d. 200 or later) all quote the verses
of Apollodorus the 'calculator' already referred to (p. 133).
But Apollodorus speaks of the ' famous theorem ', or perhaps
' figure ' (ypdfMfj.a), the discovery of which was the occa-
sion of the sacrifice, without saying what the theorem was.
Apollodorus is otherwise unknown ; he may have been earlier
than Cicero, for Cicero 5 tells the story in the same form
without specifying what geometrical discovery was meant,
and merely adds that he does not believe in the sacrifice,
because the Pythagorean ritual forbade sacrifices in which
blood was shed. Vitruvius 6 (first century B.C.) connects the
sacrifice with the discovery of the property of the particular
triangle 3, 4, 5. Plutarch, in quoting Apollodorus, questions
whether the theorem about the square of the hypotenuse was
meant, or the problem of the application of an area, while in
another place 7 he says that the occasion of the sacrifice was
1 An. Post. i. 24, 85 b 38 ; ib. ii. 17, 99 a 19.
2 Plutarch, Non posse suaviter vivi secundum Epicwum, c. 11, p. 1094 b.
3 Athenaeus x. 418 P. 4 Diog. L. viii. 12, i. 25.
5 Cicero, De nat. deor. iii. 36, 88.
6 Vitruvius, De architectural, ix. pref.
7 Plutarch, Quaest. conviv. viii. 2, 4, p. 720 A.
THE 'THEOREM OF PYTHAGORAS' 145
the solution of the problem, 'given two figures, to apply
a third which shall be equal to the one and similar to
the other '/and he adds that this problem is unquestionably
finer than the theorem about the square on the hypotenuse.
But Athenaeus and Porphyry 1 (a.d. 233-304) connect the
sacrifice with the latter proposition ; so does Diogenes Laertius
in one place. We come lastly to Proclus, who is very cautious,
mentioning the story but declining to commit himself to
the view that it was Pythagoras or even any single person
who made the discovery :
' If we listen to those who wish to recount ancient history,
we may find some of them referring this theorem to Pytha-
goras, and saying that he sacrificed an ox in honour of his
discovery. But for my part, while I admire those ivho first
observed the truth of this theorem, I marvel more at the
writer of the Elements, not only because he made it fast by a
most lucid demonstration, but because he compelled assent to
the still more general theorem by the irrefutable arguments of
science in the sixth book.'
It is possible that all these authorities may have built upon
the verses of Apollodorus ; but it is remarkable that, although
in the verses themselves the particular theorem is not speci-
fied, there is practical unanimity in attributing to Pythagoras
the theorem of Eucl. I. 47. Even in Plutarch's observations
expressing doubt about the particular occasion of the sacrifice
there is nothing to suggest that he had any hesitation in
accepting as discoveries of Pythagoras both the theorem of the
square on the hypotenuse and the problem of the application
of an area. Like Hankel, 2 therefore, I would not go so far as
to deny to Pythagoras the credit of the discovery of our pro-
position ; nay, I like to believe that tradition is right, and that
it was really his.
True, the discovery, is also claimed for India. 3 The work
relied on is the Aijastamba-feulba- Sutra, the date of which is
put at least as early as the fifth or fourth century B.C., while
it is remarked that the matter of it must have been much
1 Porphyry, Vit. Pyth. 36.
2 Hankel, Zur Geschichte der Math, in Alterthum und Mittelalter, p. 97.
3 Biirk in the Zeitschrift der morgeriland. Gesellschaft, lv, 1901,
pp. 543-91 ; lvi, 1902, pp. 327-91.
1523 L
146 PYTHAGOREAN GEOMETRY
older than the book itself ; thus one of the constructions for
right angles, using cords of lengths 15, 36, 39 (= 5, 12, 13), was
known at the time of the Tdittiriya Samhiia and the Sata-
patka Brdhmana, still older works belonging to the eighth
century B.C. at latest. A .feature of the Apastamba- tiulba-
Sutra is the construction of right angles in this way by means
of cords of lengths equal to the three sides of certain rational
right-angled triangles (or, as Apastamba calls them, rational
rectangles, i.e. those in which the diagonals as well as the
sides are rational). The rational right-angled triangles*actually
used are (3, 4, 5), (5, 12, 13), (8, 15, 17), (12, 35, 37). There is
a proposition stating tke theorem of Eucl. I. 47 as a fact in
general terms, but without proof, and there are rules based
upon it for constructing a square equal to (l) the sum of two
given squares and (2) the difference of two squares. But
certain considerations suggest doubts as to whether the
proposition had been established by any proof applicable to
all cases. Thus Apastamba mentions only seven rational
right-angled triangles, really reducible to the above-mentioned
four (one other, 7, 24, 25, appears, it is true, in the Baudha-
yana S. S., supposed to be older than Apastamba) ; he had no
general rule such as that attributed to Pythagoras for forming
any number of rational right-angled triangles; he refers to
his seven in the words ' so many recognizable constructions
are there ', implying that he knew of no other such triangles.
On the other hand, the truth of the theorem was recognized in
the case of the isosceles right-angled triangle ; there is even
a construction for V2, or the length of the diagonal of a square
with side unity, which is constructed as ( 1 -I h )
J V 3 3.4 3.4.34'
of the side, and is then used with the side for the purpose of
drawing the square on the side : the length taken is of course
an approximation to V2 derived from the consideration that
2.12 2 = 288 = 17 2 — 1 ; but the author does not say anything
which suggests any knowledge on his part that the approxi-
mate value is not exact. . Having drawn by means of the
approximate value of the diagonal an inaccurate square, he
proceeds to use it to construct a square with area equal to
three times the original square, or, in other words, to con-
struct \/3, which is therefore only approximately found.
THE ' THEOREM OF PYTHAGORAS' 147
Thus the theorem is enunciated and used as if it were of
general application ; there is, however, no sign of any general
proof ; there is nothing in fact to show that the assumption of
its universal truth was founded on anything better than an
imperfect induction from a certain number of cases, discovered
empirically, of triangles with sides in the ratios of whole
numbers in which the property (1) that the square on the
longest side is equal to the sum of the squares on the other
two was found to be always accompanied by the property
(2) that the latter two sides include a right angle. But, even
if the Indians had actually attained to a scientific proof of
the general theorem, there is no evidence or probability that
the Greeks obtained it from India ; the subject was doubtless
developed quite independently in the two countries.
The next question is, how was the theorem proved by
Pythagoras or the Pythagoreans? Vitruvius says that
Pythagoras first discovered the triangle (3, 4, 5), and doubtless
the theorem was first suggested by the discovery that this
triangle is right-angled ; but this discovery probably came
to Greece from Egypt. Then a very simple construction
would show that the theorem is true of an isosceles right-
angled triangle. Two possible lines are suggested on which
the general proof may have been developed. One is that of
decomposing square and rectangular areas into squares, rect-
angles and triangles, and piecing them together again after
the manner of Eucl., Book II ; the isosceles right-angled
triangle gives the most obvious case of this method. The
other line is one depending upon proportions ; and we have
good reason for supposing that Pythagoras developed a theory
of proportion. That theory was applicable to commensurable
magnitudes only ; but this would not be any obstacle to the
use of the method so long as the existence of the incom-
mensurable or irrational remained undiscovered. From
Proclus's remark that, while he admired those who first
noticed the truth of the theorem, he admired Euclid still
more for his most clear proof of it and for the irrefutable
demonstration of the extension of the theorem in Book VI,
it is natural to conclude that Euclid's proof in I. 47 was new,
though this is not quite certain. Now VI. 31 could be proved
at once by using I. 47 along with VI. 22 ; but Euclid proves
L 2
148 PYTHAGOREAN GEOMETRY
it independently of I. 47 by means of proportions. This
seems to suggest that he proved I. 47 by the methods of
Book I instead of by proportions in order to get the proposi-
tion into Book I instead of Book VI, to which it must have
been relegated if the proof by proportions had been used.
If, on the other hand, Pythagoras had proved it by means
of the methods of Books I and II, it would hardly have been
necessary for Euclid to devise a new proof of I. 47. Hence
it would appear most probable that Pythagoras would prove
the proposition by means of his (imperfect) theory of pro-
portions. The proof may have taken one of three different
shapes.
(1) If ABC is a triangle right-
angled at A, and AD is perpen-
dicular to BC, the triangles DBA,
DAG are both similar to the tri-
angle ABC.
It follows from the theorems of
Eucl. VI. 4 and 17 that
BA 2 = BD.BC,
AC 2 = CD.BC,
whence, by addition, BA 2 + AC 2 = BC 2 .
It will be observed that this proof is in substance identical
with that of Eucl. I. 47, the difference being that the latter
uses the relations between parallelograms and triangles on
the same base and between the same parallels instead of
proportions. The probability is that it was this particular
proof by proportions which suggested to Euclid the method
of I. 47 ; but the transformation of the proof depending on
proportions into one based on Book I only (which was abso-
lutely required under Euclid's arrangement of the Elements)
was a stroke of genius.
(2) It would be observed that, in the similar triangles
DBA, DAC, ABC, the corresponding sides opposite to the
right angle in each case are BA, AC, BC.
The triangles therefore are in the duplicate ratios of these
sides, and so are the squares on the latter.
But of the triangles two, namely DBA. DAC, make up the
third, ABC.
THE 'THEOREM OF PYTHAGORAS'
149
The same must therefore be the case with the squares, or
(3) The method of VI. 31 might have been followed
exactly, with squares taking the place of any similar recti-
lineal figures. Since the triangles DBA, ABC are similar,
BD:AB = AB:BC,
or BD, AB, BC are three proportionals, whence
AB 2 : BC 2 = BD 2 : AB 2 = BD : BC.
Similarly, A C 2 : BC 2 = CD : BC.
Therefore (BA 2 + AC 2 ) : BC 2 = (BD + DC) : BC. [V. 24]
= 1.
If, on the other hand, the proposition was originally proved
by the methods of Euclid, Books I, II alone (which, as I have
said, seems the less probable supposition), the suggestion of
4*
/
/
/
/
/
/
4 Bretschneider and Hankel seems to be the best. According
to this we are to suppose, first, a figure like that of Eucl.
II. 4, representing a larger square, of side (a + b), divided
into two smaller squares of sides a, b respectively, and
two complements, being two equal rectangles with a, b as
sides.
Then, dividing each complementary rectangle into two ,
equal triangles, we dispose the four triangles round another
square of side a + b in the manner shown in the second figure.
Deducting the four triangles from the original square in
each case we get, in the first figure, two squares a 2 and b 2
and, in the second figure, one square on c, the diagonal of the
rectangle (a, b) or the hypotenuse of the right-angled triangle
in which a, b are the sides about the right angle. It follows
that a 2 + b 2 = c 2 .
150 PYTHAGOREAN GEOMETRY
(y) Application of areas and geometrical algebra.
We have seen that, in connexion with the story of the
sacrifice of an ox, Plutarch attributes to Pythagoras himself
the discovery of the problem of the application of an area
or, as he says in another place, the problem ' Given two
figures, to " apply " a third figure which shall be equal to the
one, and similar to the other (of the given figures).' The
latter problem (= Eucl. VI. 25) is, strictly speaking, not so
much a case of applying an area as of constructing a figure,
because the base is not given in length ; but it depends
directly upon the simplest case of ' application of areas ',
namely the problem, solved in Eucl. I. 44, 45, of applying
to a given straight line as base a parallelogram containing
a given angle and equal in area to a given triangle or
rectilineal figure. The method of application of areas is
fundamental in Greek geometry and requires detailed notice.
We shall see that in its general form it is equivalent to the
geometrical solution of a mixed quadratic equation, and it is
therefore an essential part of what has been appropriately
called geometrical algebra.
It is certain that the theory of application of areas
originated with the Pythagoreans, if not with Pythagoras
himself. We have this on the authority of Eudemus, quoted
in the following passage of Proclus :
' These things, says Eudemus, are ancient, being discoveries
of the Muse of the Pythagoreans, I mean the application of
areas (irapafioXr) tg>v xoopicov), their exceeding (v7T€p/3o\rj) and
their falling short (e\\€i\jns). It was from the Pythagoreans
that later geometers [i. e. Apollonius of Perga] took the
names, which they then transferred to the so-called conic
lines (curves), calling one of these a parabola (application),
another a hyperbola (exceeding), and the third an ellipse
(falling short), whereas those god-like men of old saw the
things signified by these names in the construction, in a plane,
of areas upon a given finite straight line. For, when you
have a straight line set out, and lay the given area exactly
alongside the whole of the straight line, they say that you
apply the said area ; when, however, you make the length of
.the area greater than the straight line, it is said to exceed,
and, when you make it less, in which case after the area has
been drawn there is some part of the straight line extending
APPLICATION OF AREAS 151
beyond it, it is said to fall short. Euclid, too, in the sixth
book speaks in this way both of exceeding and falling short ;
but in this place (I. 44) he needed the application simply, as
he sought to apply to a given straight line an area equal
to a given triangle, in order that we might have in our
power, not only the construction (crva-Taoris) of a parallelogram
equal to a given triangle, but also the application of it to
a limited straight line.' l
The general form of the problem involving application
with exceeding or falling short is the following:
' To apply to a given straight line a rectangle (or, more
generally, a parallelogram) equal to a given rectilineal figure,
and (1) exceeding or (2) falling short by a square figure (or,
in the more general case, by a parallelogram similar to a given
parallelogram).'
The most general form, shown by the words in brackets,
is found in »Eucl. VI. 28, 29, which are equivalent to the
geometrical solution of the quadratic equations
ax + - x l = — 3,
— c m
and VI. 27 gives the condition of possibility of a solution
when the sign is negative and the" parallelogram falls short.
This general case of course requires the use of proportions;
but the simpler case where the area applied is a rectangle, ■
and the form of the portion which overlaps or falls short
is a square, can be,, solved by means of Book II only. The
proposition II. 1 1 is the geometrical solution of the particular
quadratic equation a (a-x)= x 2 ,
or x 2 + ax — a 2 .
The propositions II. 5 and 6 are in the form of theorems.
Taking, e.g., the figure of the former proposition, and sup-
posing AB = a, BD = x, we have
ax — x 2 = rectangle AH
= gnomon NOP.
If, then, the area of the gnomon is given ( = b 2 , say, for any
area can be transformed into the equivalent square by means
of the problems of Eucl. I. 45 and II. 14), the solution of the
equation ax-x 2 = b 2
1 Proclus on Eucl. I, pp. 419. 15-420. 12.
152
PYTHAGOREAN GEOMETRY
would be, in the language of application of areas, ' To a given
straight line (a) to apply a rectangle which shall be equal
to a given square (b~) and shall fall short by a square figure.'
A
C
R
\c
> B
/
t
'"/
\0
K
L
N>
' H
— 1 —
/
P
a
M
As the Pythagoreans solved the somewhat similar equation
in II. 1 1 , they cannot have failed to solve this one, as well as
the equations corresponding to II. 6. For in the present case
it is only necessary to draw CQ at right angles to A B from
its middle point C, to make CQ equal to h, and then, with
centre Q and radius equal to CB, or \a, to draw a circle
cutting QC produced in R and CB in D (b 2 must be not
greater than \a 2 ; otherwise a solution is impossible).
Then the determination of the point D constitutes the
solution of the quadratic.
For, by the proposition II. 5,
AD.DB + CD 2 = CB 2
= QD 2 = QC 2 + CD 2 ;
therefore AD.DB = QC 2 ,
2 = b 2 .
or
ax — x'
Similarly II. 6 enables us to solve the equations
ax + x 2 = b 2 ,
and x 2 — ax = b 2 ;
the first equation corresponding to AB = a, BD = x and the
second to AB = a, AD = x, in the figure of the proposition.
The application of the theory to conies by Apollonius will
be described when we come to deal with his treatise.
One great feature of Book II of Euclid's Elements is the
use of the gnomon (Props. 5 to 8), which is undoubtedly
Pythagorean and is connected, as we have seen, with the
APPLICATION OF AREAS 153
application of areas. The whole of Book II, with the latter
section of Book I from Prop. 42 onwards, may be said to deal
with the transformation of areas into equivalent areas of
different shape or composition by means of ' application '
and the use of the theorem of I. 47. Eucl. II. 9 and 10 are
special cases which are very useful in geometry generally, but
were also employed by the Pythagoreans for the specific purpose
of proving the property of ' side- ' and ' diameter- ' numbers,
the object of which was clearly to develop a series of closer
and closer approximations to the value of V2 (see p. 93 ante).
The geometrical algebra, therefore, as we find it in Euclid,
Books I and II, was Pythagorean. It was of course confined
to problems not involving expressions above the second degree.
Subject to this, it was an effective substitute for modern
algebra. The product of two linear factors was a rect-
angle, and Book II of Euclid made it possible to multiply
two factors with any number of linear terms in each ; the
compression of the result into a single product (rectangle)
followed by means of the abdication-theorem (Eucl. I. 44).
That theorem itself corresponds to dividing the product of
any two linear factors by a third linear expression. To trans-
form any area into a square, we have only to turn the area
into a rectangle (as in Eucl. I. 45), and then find a square
equal to that rectangle by the method of Eucl. II. 14; the
latter problem then is equivalent to the extraction of the square
root. And we have seen that the theorems of Eucl. II. 5, 6
enable mixed quadratic equations of certain types to be solved
so far as their roots are real. In cases • where a quadratic
equation has one or both roots negative, the Greeks would
transform it into one having a positive root or roots (by the
equivalent of substituting — x for x) ; thus, where one root is
positive and one negative, they would solve the problem in
two parts by taking two cases.
The other great engine of the Greek geometrical algebra,
namely the method of proportions, was not in its full extent
available to the Pythagoreans because their theory of pro-
portion was only applicable to commensurable magnitudes
(Eudoxus was the first to establish the general theory, applic-
able to commensurables and incommensurables alike, which
we find in Eucl. V, VI). Yet it cannot be doubted that they
154 PYTHAGOREAN GEOMETRY
used the method quite freely before the discovery of the irra-
tional showed them that they were building on an insecure
and inadequate foundation.
(8) The irrational.
To return to the sentence about Pythagoras in the summary
of Proclus already quoted more than once (pp. 84, 90, 141).
Even if the reading dXoyooy were right and Proclus really
meant to attribute to Pythagoras the discovery of ' the theory,
or study; of irrationals ', it would be necessary to consider the
authority for this statement, and how far it is supported by
other evidence. We note that it occurs in a relative sentence
o? 8rj ... , which has the appearance of being inserted in paren-
thesis by the compiler of the summary rather than copied from
his original source ; and the shortened form of the first part
of the same summary published in the Variae collectiones of
Hultsch's Heron, and now included by Heiberg in Heron's
Definitions, 1 contains no such parenthesis. Other authorities
attribute the discovery of the theory of the irrational not to
Pythagoras but to the Pythagoreans. A scholium to Euclid,
Book X, says that
' the Pythagoreans were the first to address themselves to the
investigation of commensurability, having discovered it as the
result of their observation of numbers ; for, while the unit is
a common measure of all numbers, they were unable to find
a common measure of all magnitudes, . . . because alL magni-
tudes are divisible ad infinitum and never leave a magnitude
which is too small to admit of further division, but that
remainder is equally divisible ad infinitum,'
and so on. The scholiast adds the legend that
' the first of the Pythagoreans who made public the investiga-
tion of these matters perished in a shipwreck '. 2
Another commentary on Eucl. X discovered by Woepcke in
an Arabic translation and believed, with good reason, to be
part of the commentary of Pappus, says that the theory of
irrational magnitudes ' had its origin in the school of Pytha-
goras '. Again, it is impossible that Pythagoras himself should
have discovered a ' theory ' or ' study ' of irrationals in any
1 Heron, vol. iv, ed. Heib., p. 108.
• 2 Euclid, ed. Heib.', vol. v, pp. 415, 417.
THE IRRATIONAL 155
proper sense. We are told in the Theaetetus x that Theodorus
of Cyrene (a pupil of Protagoras and the teacher of Plato)
proved the irrationality of a/3, V5, &c, up to Vl7, and this
must have been at a date not much, if anything, earlier than
400 b. c. ; while it was Theaetetus who, inspired by Theodorus' s
investigation of these particular 'roots' (or surds), was the
first to generalize the theory, seeking terms to cover all such
incommensurables ; this is confirmed by the continuation of
the passage from Pappus's commentary, which says that the
theory was
'considerably developed by Theaetetus the Athenian, who
gave proof, in this part of mathematics as in others, of ability
which has been justly admired ... As for the exact dis-
tinctions of the above-named magnitudes and the rigorous
demonstrations of the propositions to which this theory gives
rise, I believe that they were chiefly established by this
mathematician '.
It follows from all this that, if Pythagoras discovered any-
thing about irrationals, it was not any ' theory ' of irrationals
but, at the most, some particular case of incommensurability.
Now the passage which states that Theodorus proved that
\/3, \/o, &c. are incommensurable says nothing of a/2. The
reason is, no doubt, that the incommensurability of a/ 2 had
been proved earlier, and everything points to the probability
that this was the first case to be discovered. But, if Pytha-
goras discovered even this, it is difficult to see how the theory
that number is the essence of all existing things, or that all
things are made of number, could have held its ground for
any length of time. The evidence suggests the conclusion
that geometry developed itself for some time on the basis of
the numerical theory of proportion which was inapplicable to
any but commensurable magnitudes, and that it received an
unexpected blow later by reason of the discovery of the irra-
tional. The inconvenience of this state of things, which
involved the restriction or abandonment of the use of propor-
tions as a method pending the discovery of the generalized
theory by Eudoxus, may account for the idea of the existence
of the irrational having been kept secret, and of punishment
having overtaken the first person who divulged it.
1 Plato, Theaetetus, 147 D sq.
156 PYTHAGOREAN GEOMETRY
If then it was not Pythagoras but some Pythagorean who
discovered the irrationality of V2, at what date are we to
suppose the discovery to have been made ? A recent writer 1
on the subject holds that it was the later Pythagoreans who
made the discovery, not much before 410 B.C. It is impos-
sible, he argues, that fifty or a hundred years would elapse
between the discovery of the irrationality of a/ 2 and the like
discovery by Theodorus (about 410 or 400 B.C.) about the other
surds a/3, \/5, &c. It is difficult to meet this argument
except by the supposition that, in the interval, the thoughts
of geometers had been taken up by other famous problems,
such as the quadrature of the circle and the duplication of the
cube (itself equivalent to finding Jj*/ 2). Another argument is
based on the passage in the Laws where the Athenian stranger
speaks of the shameful ignorance of the generality of Greeks,
who are not aware that it is not all geometrical magnitudes
that are commensurable with one another ; the speaker adds
that it was only ' late ' (oyjri noTe) that he himself learnt the
truth. 2 Even if we knew for certain whether ' late ' means
' late in the day ' or ' late in life ', the expression would not
help much towards determining the date of the first discovery
of the irrationality of a/2 ; for the language of the passage is
that of rhetorical exaggeration (Plato speaks of men who are
unacquainted with the existence of the irrational as more
comparable to swine than to human beings). Moreover, the
irrational appears in the Rejjublic as something well known,
and precisely with reference to a/2; for the expressions 'the
rational diameter of (the square the side of which is) 5 '
[= the approximation a/(49) or 7] and the 'irrational
{app-qros) diameter of 5 ' [ = \/(50)] are used without any word
of explanation. 3
Further, we have a well-authenticated title of a work by
Democritus (born 470 or 460 B.C.), wepl d\6y(ov ypajijicov ical
vol(ttS>v a/3, ' two books on irrational lines and solids ' (vacrrov
is 7r\r)pes, 'full', as opposed to kzvov. 'void', and Democritus
called his ' first bodies ' vacrrd). Of the contents of this work
we are not informed ; the recent writer already mentioned
1 H. Vogfc in Bibliotheca mathematical x 3 , 1910, pp. 97-155 (cf. ix 3 ,
p. 190 sq.).
2 Plato, Laws, 819 D-820 c. 3 Plato, Republic, vii. 546 d.
THE IRRATIONAL 157
suggests that dXoyo? does not here mean irrational or incom-
mensurable at all. but that the book was an attempt to con-
nect the atomic theory with continuous magnitudes (lines)
through ' indivisible lines ' (cf. the Aristotelian treatise n
indivisible lines), and that Democritus meant to say that,
since any two lines are alike made up of an infinite number
of the (indivisible) elements, they cannot be said to have any
expressible ratio to one another, that is, he would regard them
as ' having no ratio ' ! It is, however, impossible to suppose
that a mathematician of the calibre of Democritus could have
denied that any two lines can have a ratio to one another ;
moreover, on this view, since no two straight lines would have
a ratio to one another, ctXoyoL ypafiaal would not be a class of
lines, but all lines, and the title would lose all point. But
indeed, as we shall see, it is also on other grounds inconceiv-
able that Democritus should have been an upholder of ' indi-
visible lines ' at all. I do not attach any importance to the
further argument used in support of the interpretation in
question, namely that aXoyos in the sense of ' irrational ' is
not found in any other writer before Aristotle, and that
Plato uses the words dpp-qros and dcrvufxerpo? only. The
latter statement is not even strictly true, for Plato does in
fact use the word dXoyoi specifically of ypaa\iai in the passage
of the Republic where he speaks of youths not being dXoyoi
cocrnep ypafj,p,ai, ' irrational like lines '- 1 Poor as the joke is,
it proves that dXoyoc ypa/jifiai was a recognized technical
term, and the remark looks like a sly reference to the very
treatise of Democritus of which we are speaking. I think
there is no reason to doubt that the book was on ' irrationals '
in the technical sense. We know from other sources that
Democritus was already on the track of infinitesimals in
geometry ; and nothing is more likely than that he would
write on the kindred subject of irrationals.
I see therefore no reason to doubt that the irrationality
of V2 was discovered by some Pythagorean at a date appre-
ciably earlier than that of Democritus ; and indeed the simple
proof of it indicated by Aristotle and set out in the propo-
sition interpolated at the end of Euclid's Book X seems
appropriate to an early stage in the development of geometry.
1 Plato, Republic, 534 d.
158 PYTHAGOREAN GEOMETRY
(e) The five regular solids.
The same parenthetical sentence in Proclus which attributes
to Pythagoras the discovery of the theory of irrationals
(or proportions) also states that he discovered the ' putting
together (a-va-Tacns) of the cosmic figures' (the five regular
solids). As usual, there has been controversy as to the sense
in which this phrase is to be taken, and as to the possibility
of Pythagoras having done what is attributed to him, in any
sense of the words. I do not attach importance to the
argument that, whereas Plato, presumably ' Pythagorizing ',
assigns the first four solids to the four elements, earth, fire,
air, and water, Empedocles and not Pythagoras was the
first to 'declare these four elements to be the material princi-
ples from which the universe was evolved ; nor do I think
it follows that, because the elements are four, only the first
four solids had been discovered at the time when the four
elements came to be recognized, and that the dodecahedron
must therefore have been discovered later. I see no reason
why all five should not have been discovered by the early
Pythagoreans before any question of identifying them with
the elements arose. The fragment of Philolaus, indeed, says
that
' there are five bodies in the sphere, the fire, water, earth,
and air in the sphere, and the vessel of the sphere itself
making the fifth \ 1
but as this is only to be understood of the elements in the
sphere of the universe, not of the solid figures, in accordance
with Diels's translation, it would appear that Plato in the
Timaews 2 is the earliest authority for the allocation, and
it may very well be due to Plato himself (were not the solids
called the ' Platonic figures ' ?), although put into the mouth
of a Pythagorean. At the same time, , the fact that the
Timtieus is fundamentally Pythagorean may have induced
Aetius's authority (probably Theophrastus) to conclude too
1 Stobaeus, Eel. I, proem. 3 (p. 18. 5 Wachsmuth) ; Diels, Vors. i 3 ,
p. 314. The Greek of the last phrase is kol 6 ras a-cpaipas 6\k«9, tt(vtttov,
but oXKa's is scarcely an appropriate word, and von Wilamowitz (Platon,
vol. ii, 1919, pp. 91-2) proposes o ras o-<fiaip.ts oknos, taking oXko? (which
implies 'winding') as volumen. We might then translate by 'the spherical
envelope '.
2 Timaeus, 53 c-55 c.
THE FIVE REGULAR SOLIDS 159
.hastily that ' here, too, Plato Pythagorizes ', and to say dog-
matically on the faith of this that
' Pythagoras, seeing that there are five solid figures, which
are also called the mathematical figures, says that the earth
arose from the cube, fire from the pyramid, air from the
octahedron, water from the icosahedron, and the sphere of
the universe from the dodecahedron.' 1
It may, I think, be conceded that Pythagoras or the early
Pythagoreans would hardly be able to ' construct ' the five
regular solids in the sense of a complete theoretical construc-
tion such as we find in Eucl. XIII ; and it is possible that
Theaetetus was the first to give these constructions, whether
eypayjre in Suidas's notice means that ' he was the first to
construct ' or ' to write upon the five solids so called \ But
there js no reason why the Pythagoreans should not have
' put together* the five figures in the manner in which Plato
puts them together in the Timaeus, namely, by bringing
a certain number of angles of equilateral triangles, squares,
or pentagons severally together at one point so as to make
a solid angle, and then completing all the solid angles in that
way. That the early Pythagoreans should have discovered
the five regular solids in this elementary way agrees well
with what we know of their having put angles of certain
regular figures round a point and shown that only three
kinds of such angles would fill up the space in one plane
round the point. 2 How elementary the construction still was
in Plato's hands may be inferred from the fact that he argues
that only three of the elements are transformable into one
another because only three of the solids are made from
equilateral triangles ; these triangles, when present in suffi-
cient numbers in given regular solids, can be separated again
and redistributed so as to form regular solids of a different
number of faces, as if the solids were really hollow shells
bounded by the triangular faces as planes or laminae (Aris-
totle criticizes this in De caclo, iii. 1) ! We may indeed treat
Plato's elementary method as an . indication that this was
actually the method employed by the earliest Pythagoreans.
1 Aefc. ii. 6. 5 (Vors. i 3 , p. 306. 3-7).
« 2 Proclus on Eucl. I, pp. 304. 11-305. 3.
160 PYTHAGOREAN GEOMETRY
Putting together squares three by three, forming eight
solid angles, and equilateral triangles three by three, four by
four, or five by five, forming four, six, or twelve solid angles
respectively, we readily form a cube, a tetrahedron, an octa-
hedron, or an icosahedron, but the fifth regular solid, the
dodecahedron, requires a new element, the regular pentagon.
True, if we form the angle of an icosahedron by putting
together five equilateral triangles, the bases of those triangles
when put together form a regular pentagon ; but Pythagoras
or the Pythagoreans would require a theoretical construction.
What is the evidence that the early Pythagoreans could have
constructed and did construct pentagons'? That they did
construct them seems established by the story of Hippasus,
' who was a Pythagorean but, owing to his being the first
to publish and write down the (construction of the) sphere
with (e*, from) the twelve pentagons, perished by shipwreck
for his impiety, but received credit for the discovery, whereas
it really belonged to HIM (eKeivov rod avSpos), for it is thus
that they refer to Pythagoras, and they do not call him by
his name.' l
The connexion of Hippasus's name with the subject can
hardly be an invention, and the story probably points to
a positive achievement by him, while of course the Pytha-
goreans' jealousy for the Master accounts for the reflection
upon Hippasus and the moral. Besides, there is evidence for
the very early existence of dodecahedra in actual fact. In
1885 there was discovered on Monte Loffa (Colli Euganei,
near Padua) a regular dodecahedron of Etruscan origin, which
is held to date from the first half of the first millennium b. c. 2
Again, it appears that there are extant no less than twenty-six
objects of dodecahedral form which are of Celtic origin. 3 It
may therefore be that Pythagoras or the Pythagoreans had
seen dodecahedra of this kind, and that their merit was to
have treated them as mathematical objects and brought
them into their theoretical geometry. Could they then have
1 Iambi. Vit. Pyth. 88, de c. math, sclent, c. 25, p. 77. 18-24.
2 F. Lindemann, ' Zur Greschichte der Polyeder und der Zahlzeichen '
(Sitzmigsber. der K. Bay. Akad. der Wiss. xxvi. 1897, pp. 625-768).
3 L. Hugo in Comptes rendus of the Paris Acad, of Sciences, lxiii, 1873,
pp. 420-1 ; lxvii, 1875, pp. 433, 472 ; lxxxi, 1879, p. 332.
THE FIVE REGULAR SOLIDS 161
constructed the regular pentagon ? The answer must, I think,
be yes. If ABODE be a regular pentagon, and AC, AD, CE
be joined, it is easy to prove, from the (Pythagorean) proposi-
tions about the sum of the internal angles of a polygon and
the sum of the angles of a triangle, that each of the angles
BAC, DAE, ECD is fths of a right angle, whence, in the
triangle A CD, the angle CAD is fths of a right angle, and
each of the base angles ACD, ADC is fths of a right angle
or double of the vertical angle CAD; and from these facts
it easily follows that, if CE and AD meet in F, CDF is an
isosceles triangle equiangular, and therefore similar, to ACD,
and also that AF = FC = CD. Now, since the triangles
ACD, CDF are similar,
AC:CD = CD:DF,
or AD:AF= AF-.FD;
that is, if AD is given, the length of AF, or CD, is found by
dividing AD at i^in * extreme and mean ratio ' by Eucl. II. 11.
This last problem is a particular case of the problem of
1 application of areas ', and therefore was obviously within
the power of the Pythagoreans. This method of constructing
a pentagon is, of course, that taught in Eucl. IV. 10, 11. If
further evidence is wanted of the interest of the early Pytha-
goreans in the regular pentagon, it is furnished by the fact,
attested by Lucian and the scholiast to the Clouds of Aristo-
phanes, that the c triple interwoven triangle, the pentagon ',
i. e. the star-pentagon, was used by the Pythagoreans as a
symbol of recognition between the members of the same school,
and was called by them Health. 1 Now it will be seen from the
separate diagram of the star-pentagon above that it actually
1 Lucian, Pro lapsu in saint. § 5 (vol. i, pp. 447-8, Jacobitz) ; schol. on
Clouds 609.
1523 M *,
162 PYTHAGOREAN GEOMETRY
shows the equal sides of the five isosceles triangles of the type
referred to and also the points at which they are divided in
extreme and mean ratio. (I should perhaps add that the
pentagram is said to be found on the vase of Aristonophus
found at Caere and supposed to belong to the seventh
century B.C., while the finds at Mycenae include ornaments of
pentagonal form.)
It would be easy to conclude that the dodecahedron is in-
seribable in a sphere, and to find the centre of it, without
constructing both in the elaborate manner of Eucl. XIII. 17
and working out the relation between an edge of the dodeca-
hedron and the radius of the sphere, as is there done : an
investigation probably due to Theaetetus. It is right to
mention here the remark in scholium No. 1 to Eucl. XIII
that the book is about
'the five so-called Platonic figures, which, however, do not
belong to Plato, three of the five being due to the Pytha-
goreans, namely the cube, the pyramid, and the dodeca-
hedron, while the .octahedron and icosahedron are due to
Theaetetus V
This statement (taken probably from Geminus) may per-
haps rest on the fact that Theaetetus was the first to write
at any length about the two last-mentioned solids, as he was
probably the first to construct all iive theoretically and in-
vestigate fully their relations to one another and the circum-
scribing spheres.
(C) Pythagorean astronomy.
Pythagoras and the Pythagoreans occupy an important place
in the history of astronomy. (1) Pythagoras was one of the first
to maintain that the universe and the earth are spherical
in form. It is uncertain what led Pythagoras to conclude
that the earth is a sphere. One suggestion is that he inferred
it from the roundness of the shadow cast by the earth in
eclipses of the moon. But it is certain that Anaxagoras was
the first to suggest this, the true, explanation of eclipses.
The most likely supposition is that Pythagoras's ground was
purely mathematical, or mathematico-aesthetical ; that is, he
1 Heiberg's Euclid, vol. v, p. 654.
PYTHAGOREAN ASTRONOMY 163
attributed spherical shape to the earth (as to the universe)
for the simple reason that the sphere is the most beautiful
of solid figures. For the same reason Pythagoras would
surely hold that the sun, the moon, and the other heavenly
bodies are also spherical in shape. (2) Pythagoras is credited
with having observed the identity of the Morning and the
Evening Stars. (3) It is probable that he was the first to
state the view (attributed to Alcmaeon and ' some of the
mathematicians') that the planets as well as the sun and
moon have a motion of their own from west to east opposite
to and independent of the daily rotation of the sphere of the
fixed stars from east to west. 1 Hermesianax, one of the older
generation of Alexandrine poets (about 300 B.C.), is quoted as
saying :
c What inspiration laid forceful hold on Pythagoras when
he discovered the subtle geometry of (the heavenly) spirals
and compressed in a small sphere the whole of the circle which
the aether embraces.' 2
This would seem to imply the construction of a sphere,
on which were represented the circles described by the sun,
moon and planets together with the daily revolution of the
heavenly sphere ; but of course Hermesianax is not altogether
a trustworthy authority.
It is improbable that Pythagoras himself was responsible
for the astronomical system known as the Pythagorean, in
which the earth was deposed from its place at rest in the
centre of the universe, and became a 'planet', like the sun,
the moon and the other planets, revolving about the central
fire. For Pythagoras the earth was still at the centre, while
about it there moved (a) the sphere of the fixed stars revolv-
ing daily from east to west, the axis of rotation being a
straight line through the centre of the earth, (b) the sun,
moon and planets moving in independent circular orbits in
a sense opposite to that of the daily rotation, i. e. from west
to east.
The later Pythagorean system is attributed by Aetius
(probably on the authority of Theophrastus) to Philolaus, and
1 Aet. ii. 16. 2, 3 {Vors. i 3 , p. 132. 15).
2 See Athenaeus, xiii. 599 a,
M 2
164 PYTHAGOREAN GEOMETRY
may be described thus. The universe is spherical in shape
and finite in size. Outside it is infinite void which enables
the universe to breathe, as it were. At the centre is the
central fire, the Hearth of the Universe, called by various
names, the Tower or Watch-tower of Zeus, the Throne of
Zeus, the House of Zeus, the Mother of the Gods, the Altar,
Bond and Measure of Nature. In this central fire is located
the governing principle, the force which directs the movement
and activity of the universe. In the universe there revolve
in circles about the central fire the following bodies. Nearest
to the central fire revolves the counter-earth, which always
accompanies the earth, the orbit of the earth coming next* to
that of the counter- earth ; next to the earth, reckoning in
order from the centre outwards, comes the moon, next to the
moon the sun, next to the sun the five planets, and last of
all, outside the orbits of the five planets, the sphere of the
fixed stars. The counter-earth, which accompanies the earth
and revolves in a smaller orbit, is not seen by us because
the hemisphere of the earth on which we live is turned away
from the counter- earth (the analogy of the moon which
always turns one side towards us may have suggested this) ;
this involves, incidentally, a rotation of the earth about its
axis completed in the same time as it takes the earth to
complete a revolution about the central fire. As the latter
revolution of the earth was held to produce day and night,
it is a natural inference that the earth was supposed to
complete one revolution round the central fire in a day and
a night, or in twenty-four hours. This motion on the part of
the earth with our hemisphere always turned outwards would,
of course, be equivalent, as an explanation of phenomena,
to a rotation of the earth about a fixed axis, but for the
parallax consequent on the earth describing a circle in space
with radius greater than its own radius ; this parallax, if we
may trust Aristotle, 1 the Pythagoreans boldly asserted to be
negligible. The superfluous thing in this system is the
introduction of the counter- earth. Aristotle says in one
place that its object was to bring up the number of the
moving bodies to ten, the perfect number according to
1 Arist. Be caelo, ii. 13, 293 b 25-30.
PYTHAGOREAN ASTRONOMY 165
•
the Pythagoreans ! ; but he hints at the truer explanation in
another passage where he says that eclipses of the moon
were considered to be due sometimes to the interposition
of the earth, sometimes to the interposition of the counter-
earth (to say nothing of other bodies of the same sort
assumed by ' some ' in order to explain why there appear
to be more lunar eclipses than solar) 2 ; we may therefore
take it that the counter-earth was invented for the purpose
of explaining eclipses of the moon and their frequency.
Recapitulation.
The astronomical systems of Pythagoras and the Pytha-
goreans illustrate the purely mathematical character of their
physical speculations; the heavenly bodies are all spheres,
the most perfect of solid figures, and they move in circles ;
there is no question raised of forces causing the respective
movements ; astronomy is pure mathematics, it is geometry,
combined with arithmetic and harmony. The capital dis-
covery by Pythagoras of the dependence of musical intervals
on numerical proportions led, with his successors, to the
doctrine of the ' harmony of the spheres '. As the ratio
2 : 1 between the lengths of strings of the same substance
and at the same tension corresponds to the octave, the
ratio 3 : 2 to the fifth, and the ratio 4 : 3 to the fourth, it
was held that bodies moving in space produce sounds, that
those which move more quickly give a higher note than those
which move more slowly, while those move most quickly which
move at the greatest distance ; the sounds therefore pro-
duced by the heavenly bodies, depending on their distances
(i.e. the size of their orbits), combine to produce a harmony;
' the whole heaven is number and harmony \ 3
We have seen too how, with the Pythagoreans, the theory
of numbers, or ' arithmetic ', goes hand in hand with geometry;
numbers are represented by dots or lines forming geometrical
figures ; the species of numbers often take their names from
their geometrical analogues, while their properties are proved
by geometry. The Pythagorean mathematics, therefore, is all
one science, and their science is all mathematics.
1 Arist. Metaph. A. 5, 986 a 8-12.
2 Arist. Be caelo, ii. 13, 293 b 21-5. 3 Arist. Metaph. A. 5, 986 a 2.
166 PYTHAGOREAN GEOMETRY
It is this identification of mathematics (and of geometry
in particular) with science in general, and their pursuit of it
for its own sake, which led to the extraordinary advance of
the subject in the Pythagorean school. It was the great merit
of Pythagoras himself (apart from any particular geometrical
or arithmetical theorems which he discovered) that he was the
first to take this view of mathematics ; it is characteristic of
him that, as we are told, ' geometry was called by Pythagoras
inquiry or science ' (eKaXeiro 8e 77 yeeo/ieTpia wpbs HvBayopov
lo-Topia). 1 Not only did he make geometry a liberal educa-
tion ; he was the first to attempt to explore it down to its
first principles ; as part of the scientific basis which he sought
to lay down he ' used definitions '. A point was, according to
the Pythagoreans, a ' unit having position ' 2 ; and, if their
method of regarding a line, a surface, a solid, and an angle
does not amount to a definition, it at least shows that they
had reached a clear idea of the differentiae, as when they said
that 1 was a point, 2 a line, 3 a triangle, and 4 a pyramid,
A surface they called xpoid, ' colour ' ; this was their way of
describing the superficial appearance, the idea being, as
Aristotle says, that the colour is either in the limiting surface
(yrepas) or is the irepas? so that the meaning intended to be
conveyed is precisely that intended by Euclid's definition
(XI. Def. 2) that ' the limit of a solid is a surface '. An angle
they called y\<o\h, a ' point ' (as of an arrow) made by a line
broken or bent back at one point. 4
The positive achievements of the Pythagorean school in
geometry, and the immense advance made by them, will be
seen from the following summary.
1, They were acquainted with the properties of parallel
lines, which they used for the purpose of establishing by
a general proof the proposition that the sum of the three
angles of any triangle is equal to two right angles. This
latter proposition they again used to establish the well-known
theorems about the sums of the exterior and interior angles,
respectively, of any polygon. *
2. They originated the subject of equivalent areas, the
transformation of an area of one form into another of different
1 Iambi. Vit. Pi/th. 89. 2 Proclus on Eucl. I, p. 95. 21.
3 Arist. De sensu, 3 ; 439 a 31. 4 Hevon, Def. 15.
RECAPITULATION 167
form and, in particular, the whole method of application of
areas, constituting a geometrical algebra, whereby they effected
the equivalent of the algebraical processes of addition, sub-
traction, multiplication, division, squaring, extraction of the
square root, and finally the complete solution of the mixed
quadratic equation x 2 ±px±q = 0, so far as its roots are real.
Expressed in terms of Euclid, this means the whole content of
Book I. 35-48 and Book II. The method of application of
areas is one of the most fundamental in the whole of later
Greek geometry ; it takes its place by the side of the powerful
method of proportions; moreover, it is the starting point of
Apollonius's theory of conies, and the three fundamental
terms, parabole, ellipsis, and hyijerbole used to describe the
three separate problems in 'application' were actually em-
ployed by Apollonius to denote the three conies, names
which, of course, are those which we use to-day. Nor was
the use of the geometrical algebra for solving numerical
problems unknown to the Pythagoreans ; this is proved by
the fact that the theorems of Eucl. II. 9, 10 were invented
for the purpose of' finding successive integral solutions of the
indeterminate equations
2x 2 -y* = ±1.
3. They had a theory of proportion pretty fully developed.
We know nothing of the form in which it was expounded;
all we know is that it took no account of incommensurable
magnitudes. Hence we conclude that it was a numerical
theory, a theory on the same lines as that contained in
Book VII of Euclid's Elements.
They were aware of the properties of similar figures.
This is clear from the fact that they must be assumed
to have solved the problem, which was, according to
Plutarch, attributed to Pythagoras himself, of describing a
figure which shall be similar to one given figure and equal in
area to another given figure This implies a knowledge of
the proposition that similar figures (triangles or polygons) are
to one another in the duplicate ratio of corresponding sides
(Eucl. VI. 19, 20). As the problem is solved in Eucl. VI. 25,
we assume that, subject to the qualification 'that their
theorems about similarity, &c, were only established of figures
168 PYTHAGOREAN GEOMETRY
in which corresponding elements are commensurable, they had
theorems corresponding to a great part of EucL, Book VI.
Again, they knew how to cut a straight line in extreme and
mean ratio (Eucl. VI. 30) ; this problem was presumably
solved by the method used in Eucl. II. 11, rather than by that
of Eucl. VI. 30, which depends on the solution of a problem
in the application of areas more general than the methods of
Book K enable us to solve, the problem namely of Eucl.
VI. 29.
4. They had discovered, or were aware of the existence of,
the five regular solids. These they may have constructed
empirically by putting together squares, equilateral triangles,
and pentagons. This implies that they could construct a
regular pentagon and, as this construction depends upon the
construction of an isosceles triangle in which each of the base
angles is double of the vertical angle, and this again on the
cutting of a line in extreme and mean ratio, we may fairly
assume that this was the way in which the construction of
the regular pentagon was actually evolved. It would follow
that the solution of problems by analysis was already prac-
tised by the Pythagoreans, notwithstanding that the discovery
of the analytical method is attributed by Proclus to Plato.
As the particular construction is practically given in Eucl. IV.
10, 11, we may assume that the content of Eucl. IV was also
partly Pythagorean.
5. They discovered the existence of the irrational in the
sense that they proved the incommensurability of the diagonal
of a square with reference to its side ; in other words, they
proved the irrationality of */2. As a proof of this is referred
to b}^ Aristotle in terms which correspond to the method
used in a proposition interpolated in Euclid, Book X, we
may conclude that this proof is ancient, and therefore that it
was probably the proof used by the discoverers of the proposi-
tion. The method is to prove that, if the diagonal of a square
is commensurable with the side, then the same number must
be both odd and even ; here then we probably have an early
Pythagorean use of the method of reductio ad absurdum.
Not only did the Pythagoreans discover the irrationality
of \^2 ; they showed, as we have seen, how to approximate
as closely as we please to its numerical value.
RECAPITULATION 169
After the discovery of this one case of irrationality, it
would be obvious that propositions theretofore proved by
means of the numerical theory of proportion, which was
inapplicable to incommensurable magnitudes, were only par-
tially proved. Accordingly, pending the discovery of a theory
of proportion applicable to incommensurable as well as com-
mensurable magnitudes, there would be an inducement to
substitute, where possible, for proofs employing the theory of
proportions other proofs independent of that theory. This
substitution is carried rather far in Euclid, Books I-IV ; it
does not follow that the Pythagoreans remodelled their proofs
to the same extent as Euclid felt bound to do.
VI
PROGRESS IN THE ELEMENTS DOWN TO
PLATO'S TIME
In tracing the further progress in the Elements which took
place down to the time of Plato, we do not get much assistance
from the summary of Proclus. The passage in which he
states the succession of geometers from Pythagoras to Plato
and his contemporaries runs as follows :
' After him [Pythagoras] Anaxagoras of Clazomenae dealt
with many questions in geometry, and so did Oenopides of
Chios, who was a little younger than Anaxagoras; Plato
himself alludes, in the Rivals, to both of them as having
acquired a reputation for mathematics. After them came
Hippocrates of Chios, the discoverer of the quadrature of
the lune, and Theodorus of Cyrene, both of whom became
distinguished geometers; Hippocrates indeed was the first
of whom it is recorded that he actually compiled Elements.
Plato, who came next to them, caused mathematics in general
and geometry in particular to make a very great advance,
owing to his own zeal for these studies; for every one knows
that he even filled his writings with mathematical discourses
and strove on every occasion to arouse enthusiasm for mathe-
matics in those who took up philosophy. At this time too
lived Leodamas of Thasos, Archytas of Taras, and Theaetetus
of Athens, by whom the number of theorems was increased
and a further advance was made towards a more scientific
grouping of them.' 1
It will be seen that we have here little more than a list of
names of persons who advanced, or were distinguished in,
geometry. There is no mention of specific discoveries made
by particular geometers, except that the work of Hippocrates
on the squaring of certain lunes is incidentally alluded to,
rather as a means of identifying Hippocrates than as a de-
tail relevant to the subject in hand. It would appear that
1 Proclus on Eucl. I, p. 65. 21-66. 18.
THE ELEMENTS DOWN TO PLATO'S TIME 171
the whole summary was directed to the one object of trac-
ing progress in the Elements, particularly with reference
to improvements of method in the direction of greater
generality and more scientific order and treatment ; hence
only those writers are here mentioned who contributed to this
development. Hippocrates comes into the list, not because
of his lunes, but because he was a distinguished geometer
and was the first to write Elements. Hippias of Elis, on the
other hand, though he belongs to the period covered by the
extract, is omitted, *presumably because his great discovery,
that of the curve known as the quadratrix, does not belong
to elementary geometry ; Hippias is, however, mentioned in
two other places by Proclus in connexion with the quadratrix, 1
and once more as authority for the geometrical achievements
of Ameristus (or Mamercus or Mamertius). 2 Less justice is
done to Democritus, who is neither mentioned here nor else-
where in the commentary ; the omission here of the name
of Democritus is one of the arguments for the view that
this part of the summary is not quoted from the History
of Geometry by Eudemus (who would not have been likely to
omit so accomplished a mathematician as Democritus), but
is the work either of an intermediary or of Proclus himself,
based indeed upon data from Eudemus's history, but limited to
particulars relevant to the object of the commentary, that
is to say, the elucidation of Euclid and the story of the growth
of the Elements.
There are, it is true, elsewhere in Proclus's commentary
a very few cases in which particular propositions in Euclid,
Book I, are attributed to individual geometers, e.g. those
which Thales is said to have discovered. Two propositions
presently to be mentioned are in like manner put to the
account of Oenopides ; but except for these details about
Oenopides we have to look elsewhere for evidence of the
growth of the Elements, in the period now under notice.
Fortunately we possess a document of capital importance,
from this point of view, in the fragment of Eudemus on
Hippocrates's quadrature of lunes preserved in Simplicius's
commentary on the Physics of Aristotle. 3 This fragment will
1 Proclus on Eucl. I, p. 272. 7, p. 356. 11. 2 lb., p. 65. 14.
3 Simpl. in Arist. Phys. pp. 54-69 Diels.
172 THE ELEMENTS DOWN TO PLATO'S TIME
be described below. Meantime we will take the names men-
tioned by Proclus in their order.
Anaxagoras (about 500-428 B.C.) was born at Clazomenae
in the neighbourhood of Smyrna. He neglected his posses-
sions, which were considerable, in order to devote himself
to science. Some one once asked him what was the object
of being born, to which he replied, ' The investigation of sun,
moon and heaven.' He was apparently the first philosopher
to take up his abode at Athens, where he enjoyed the friend-
ship of Pericles. When Pericles became unpopular shortly
before the outbreak of the Peloponnesian War, he was attacked
through his friends, and Anaxagoras was accused of impiety
for holding that the sun was a red-hot stone and the moon
earth. According to one account he was fined five talents
and banished ; another account says that he was kept in
prison and that it was intended to put him to death, but
that Pericles obtained his release ; he went and lived at
Lampsacus till his death.
Little or nothing is known of Anaxagoras's achievements
in mathematics proper, though it is credible enough that
he was a good mathematician. But in astronomy he made
one epoch-making discovery, besides putting forward some
remarkably original theories about the evolution of the'
universe. We owe to him the first clear recognition of the
fact that the moon does not shine by its own light but
receives its light from the sun ; this discovery enabled him
to give the true explanation of lunar and solar eclipses,
though as regards the former (perhaps in order to explain
their greater frequency) he erroneously supposed that there
were other opaque and invisible bodies ' below the moon '
which, as well as the earth, sometimes by their interposition
caused eclipses of the moon. A word should be added about
his cosmology on account of the fruitful ideas which it con-
tained. According to him the formation of the world began
with a vortex set up, in a portion of the mixed mass in which
'all things were together', by Mind (uov?). This rotatory
movement began in the centre and then gradually spread,
taking in wider and wider circles. The first effect was to
separate two great masses, one consisting of the rare, hot,
light, dry, called the 'aether', the other of the opposite
ANAXAGORAS 173
categories and called 'air'. The aether took the outer, the
air the inner place. From the air were next separated clouds,
water, earth and stones. The dense, the moist, the dark and
cold, and all the heaviest things, collected in the centre as the
result of the circular motion, and it was from these elements
when consolidated that the earth was formed ; but after this,
in consequence of the violence of the whirling motion, the
surrounding fiery aether tore stones away from the earth and
kindled thern into stars. Taking this in conjunction with
the remark that stones 'rush outwards more than water \
we see that Anaxagoras conceived the idea of a centrifugal
force as well as that of concentration brought about by the
motion of the vortex, and that he assumed a series of pro-
jections or ' whirlings-off' of precisely the same kind as the
theory of Kant and Laplace assumed for the formation of
the solar system. At the same time he held that one of the
heavenly bodies might break away and fall (this may account
for the story that he prophesied the fall of the meteoric stone
at Aegospotami in 468/7 B.C.), a centripetal tendency being
here recognized.
In mathematics we are told that Anaxagoras 'while in
prison wrote (or drew, eypacpe) the squaring of the circle \ l
But we have no means of judging what this amounted to.
Rudio translates eypafa as ' zeichnete ', 'drew ', observing that
he probably knew the Egyptian rule for squaring, and simply
drew on the sand a square as nearly as he could equal to the
area'of a circle. 2 It is clear to me that this cannot be right,
but that the word means ' wrote upon ' in the sense that he
tried to work out theoretically the problem in question. For
the same word occurs (in the passive) in the extract from
Eudemus about Hippocrates : ' The squarings of the lunes . . .
were first written (or proved) by Hippocrates and were found
to be correctly expounded', 3 where the context shows that
kypdcp-qcrav cannot merely mean 'were drawn'. Besides,
TeTpaycovio-fios, squaring, is a process or operation, and you
cannot, properly speaking, ' draw ' a process, though you can
' describe ' it or prove its correctness.
1 Plutarch, De exil. 17, 607 f.
2 Rudio, Der BericJit des Simplicius ilber die Quadrat uren des Antiphon
und Hippokrates, 1907, p. 92, 93.
3 Simpl. in Phys., p. 61. 1-3 Diels ; Rudio, op. cit., pp. 46. 22-48. 4.
174 THE ELEMENTS DOWN TO PLATO'S TIME
Vitruvius tells us that one Agatharchus was the first to paint
stage-scenes at Athens, at the time when Aeschylus was
having his tragedies performed, and that he left a treatise on
the subject which was afterwards a guide to Democritus and
Anaxagoras, who discussed the same problem, namely that of
painting objects on a plane surface in such a way as to make
some of the things depicted appear to be in the background
while others appeared to stand out in the foreground, so that
you seemed, e.g., to have real buildings before you ; in other
words, Anaxagoras and Democritus both wrote treatises on
perspective. 1
There is not much to be gathered from the passage in
the Rivals to which Proclus refers. Socrates, on entering the
school of Dionysius, finds two lads disputing a certain point,
something about Anaxagoras or Oenopides, he was not certain
which ; but they appeared to be drawing circles, and to be
imitating certain inclinations by placing their hands at an
angle. 2 Now this description suggests that what the lads
were trying to represent was the circles of the equator and
the zodiac or ecliptic ; and we know that in fact Eudemus
in his History of Astronomy attributed to Oenopides th© dis-
covery of ' the cincture of the zodiac circle ', 3 which must mean
the discovery of the obliquity of the ecliptic. It would prob-
ably be unsafe to conclude that Anaxagoras was also credited
with the same discovery , but it certainly seems to be suggested
that Anaxagoras had to some extent touched the mathematics
of astronomy.
Oenopides of Chios was primarily an astronomer. This
is shown not only by the reference of Eudemus just cited, but
by a remark of Proclus in connexion with one of two proposi-
tions in elementary geometry attributed to him. 4 Eudemus
is quoted as saying that he not only discovered the obliquity
of the ecliptic, but also the period of a Great Year. Accord-
ing to Diodorus the Egyptian priests claimed that it was from
them that Oenopides learned that the sun moves in an inclined
orbit and in a sense opposite to the motion of the fixed stars.
It does not appear that Oenopides made any measurement of
1 Vitruvius, De architecture, vii. praef. 11.
2 Plato, Erastae 132 a, b. 3 Theon of Smyrna, p. 198. 14.
4 Proclus on Eucl. I, p. 283. 7-8.
OENOPIDES OF CHIOS 175
the obliquity of the ecliptic. The duration of the Great Year
he is said to have put at 59 years, while he made the length
of the year itself to be 365f § days. His Great Year clearly
had reference to the sun and moon only ; he merely sought to
find the least integral number of complete years which would
contain an exact number of lunar months. Starting, probably,
with 365 days as the length of a year and 29 J days as the
length of a lunar month, approximate values known before
his time, he would see that twice 2 9 J, or 59, years would con-
tain twice 365, or 730, lunar months. He may then, from his
knowledge of the calendar, have obtained 21,557 as the num-
ber of days in 730 months, for 21,557 when divided by 59 gives
365-|| as the number of days in the year.
Of Oenopides's geometry we have no details, except that
Proclus attributes to him two propositions in Eucl. Bk. I. Of
I. 1 2 (' to draw a perpendicular to a given straight line from
a point outside it ') Proclus says :
'This problem was first investigated by Oenopides, who
thought it useful for astronomy. He, however, calls the per-
pendicular in the archaic manner (a straight line drawn)
gnomon-wise (Kara yv<Z>\iovci), because the gnomon is also at
right angles to the horizon.' *
On I. 23 (' on a given straight line and at a given point on
it to construct a rectilineal angle equal to a given rectilineal
angle ') Proclus remarks that this problem is ' rather the dis-
covery of Oenopides, as Eudemus says \ 2 It is clear that the
geometrical reputation of Oenopides could not have rested on
the mere solution of such simple problems as these. Nor, of
course, could he have been the first to draw a perpendicular in
practice ; the point may be that he was the first to solve the
problem by means of the ruler and compasses only, whereas
presumably, in earlier days, perpendiculars would be drawn
by means of a set square or a right-angled triangle originally
constructed, say, with sides proportional to 3, 4, 5. Similarly
Oenopides may have been the first to give the theoretical,
rather than the practical, construction for the problem of I. 23
which we find in Euclid. It may therefore be that Oenopides's
significance lay in improvements of method from the point of
view of theory ; he may, for example, have been the first to
1 Proclus on Eucl. I, p. 283. 7-8. 2 Proclus on Eucl. I, p. 333. 5.
176 THE ELEMENTS DOWN TO PLATO'S TIME
lay down the restriction of the means permissible in construc-
tions to the ruler and compasses which became a canon of
Greek geometry for all 'plane' constructions, i.e. for all
problems involving the equivalent of the solution of algebraical
equations of degree not higher than the second.
Democritus, as mathematician, may be said to have at last
come into his own. In the Method of Archimedes, happily
discovered in 1906, we are told that Democritus was the first
to state the important propositions that the volume of a cone
is one third of that of a cylinder having the same base and
equal height, and that the volume of a pyramid is one third of
that of a prism having the same base and equal height ; that is
to say, Democritus enunciated these propositions some fifty
years or more before they were first scientifically proved by
Eudoxus.
Democritus came from Abdera, and, according to his own
account, was young when Anaxagoras was old. Apollodorus
placed his birth in 01. 80 (= 460-457 B.C.), while according
to Thrasyllus he was born in 01. 77. 3 (= 470/69 B.C.), being
one year older than Socrates. He lived to a great age, 90
according to Diodorus, 104, 108, 109 according to other
authorities. He was indeed, as Thrasyllus called him,
irzvTaQXos in philosophy 1 ; there was no subject to which he
did not notably contribute, from mathematics and physics on
the one hand to ethics and poetics on the other ; he even went
by the name of ' Wisdom ' (%o(j)ia). 2 Plato, of course, ignores
him throughout his dialogues, and is said to have wished to
burn all his works; Aristotle, on the other hand, pays
handsome tribute to his genius, observing, e.g., that on the
subject of change and growth no one save Democritus had
observed anything except superficially; whereas Democritus
seemed to have thought of everything. 3 He could say
of himself (the fragment is, it is true, considered by Diels
to be spurious, while Gomperz held it to be genuine), ' Of
all my contemporaries I have covered the most ground in
my travels, making the most exhaustive inquiries the while ;
I have seen the most climates and countries and listened to
1 Diog. L. ix. 37 (Vors. ii 3 , p. 11. 24-30).
2 Clem. Strom, vi. 32 (Vors. ii 3 , p. 16. 28).
3 Arist. De gen. et corr. i. 2, 315 a 35.
DEMOCRITUS 177
the greatest number of learned men \ 1 His travels lasted for
rive years, and he is said to have visited Egypt, Persia and
Babylon, where he consorted with the priests and magi ; some
gay that he went to India and Aethiopia also. Well might
he undertake the compilation of a geographical survey of
the earth as, after Anaximander, Hecataeus of Miletus and
Damastes of Sigeum had done. In his lifetime his fame was
far from world-wide : ' I came to Athens ', he says, ' and no
one knew me.' 2
A long lif=t of his writings is preserved in Diogenes Laertius,
the authority being Thrasyllus. In astronomy he wrote,
among other works, a book On the Planets, and another On
the Great Year or Astronomy including a parapegma 3 (or
calendar). Democritus made the order of the heavenly bodies,
reckoning outwards from the earth, the following: Moon,
Venus, Sun, the other planets, the fixed stars. Lucretius 4 has
preserved an interesting explanation which he gave of the
reason why the sun takes a year to describe the full circle of
the zodiac, while the moon completes its circle in a month.
The nearer any body is to the earth (and therefore the farther
from the sphere of the fixed stars) the less swiftly can it be
carried round by the revolution of the heaven. Now the
moon is nearer than the sun, and the sun than the signs of
the zodiac ; therefore the moon seems to get round faster than
the sun because, while the sun, being lower and therefore
slower than the signs, is left behind by them, the moon,
being still lower and therefore slower still, is still more left
behind. Democritus's Great Year is described by Censorinus 5
as 82 (LXXXII) years including 28 intercalary months, the
latter number being the same as that included by Callippus in
his cycle of 76 years ; it is therefore probable that LXXXII
is an incorrect reading for LXXVII (77).
As regards his mathematics we have first the statement in
1 Clement, Strom, i. 15, 69 (Vors. ii 3 , p. 123. 3).
2 Diog. L. ix. 36 (Vors. ii 3 , p. 11. 22).
3 The parapegma was a posted record, a kind of almanac, giving, for
a series of years, the movements of the sun, the dates of the phases of
the moon, the risings and settings of certain stars, besides f'7rto-^/Ltao-mi
or weather indications ; many details from Democritus's parapegma
are preserved in the Calendar at the end of Geminus's Isagoge and in
Ptolemy.
4 Lucretius, v. 621 sqq. 5 De die nataN, 18. 8.
1523 N
178 THE ELEMENTS DOWN TO PLATO'S TIME
the continuation of the fragment of doubtful authenticity
already quoted that
'in the putting together of lines, with the necessary proof, no
one has yet surpassed me, not even the so-called harpedon-
aptae (rope-stretchers) of Egypt '.
This does not tell us much, except that it indicates that
the ' rope-stretchers ', whose original function was land-
measuring or practical geometry, had by Democritus's time
advanced some way in theoretical geometry (a fact which the
surviving documents, such as the book of Ahmes, with their
merely practical rules, would not have enabled us to infer).
However, there is no reasonable doubt that in geometry
Democritus was fully abreast of the knowledge of his day ;
this is fully confirmed by the titles of treatises by him and
from other sources. The titles of the works classed as mathe-
matical are (besides the astronomical works above mentioned) :
1. On a difference of opinion (yvcourjs: v. I. yv&uovos, gno-
mon), or on the contact of a circle and a sphere;
2. On Geometry ;
3 . Geometricorum (? I, II) ;
4. Numbers;
5. On irrational lines and solids (vcccttcoi', atoms ?);
6. 'EKTrerdariiaTa.
As regards the first of these works I think that the
attempts to extract a sense out of Cobet's reading yv&uovo?
(on a difference of a gnomon) have failed, and that yvdofir}?
(Diels) is better. But ' On a difference of opinion ' seems
scarcely determinative enough, if this was really an alternative
title to the book. We know that there were controversies in
ancient times about the nature of the ' angle of contact ' (the
' angle ' formed, at the point of contact, between an arc of
a circle and the tangent to it, which angle was called by the
special name hornlike, KeparoeiBfjs), and the 'angle' comple-
mentary to it (the 'angle of a semicircle 'J. 1 The question was
whether the ' hornlike angle ' was a magnitude comparable
with the rectilineal angle, i.e. whether by being multiplied
a sufficient number of times it could be made to exceed a
1 Proclus on Eucl. I, pp. 121. 24-122- 6.
DEMOCRITUS 179
given rectilineal angle. Euclid proved (in III. 16) that the
■ angle of contact ' is less than any rectilineal angle, thereby
setting the question at rest. This is the only reference in
Euclid to this angle and the ' angle of a semicircle ', although
he demies the 'angle of a segment' in III, Def. 7, and has
statements about the angles of segments in III. 31. But we
know from a passage of Aristotle that before his time ' angles
of segments ' came into geometrical text-books as elements in
figures which could be used in the proofs of propositions ] ;
thus e.g. the equality of the two angles of a segment
(assumed as known) was used to prove the theorem of
Eucl. I. 5. Euclid abandoned the use of all such angles in
proofs, and the references to them above mentioned are only
survivals. The controversies doubtless arose long before his
time, and such a question as the nature of the contact of
a circle with its tangent would probably have a fascination
for Democritus, who, as we shall see, broached other questions
involving infinitesimals. As, therefore, the questions of the
nature of the contact of a circle with its tangent and of the
character of the ' hornlike ' angle are obviously connected,
I prefer to read ycovi-qs (' of an angle ') instead of yucofir]^ ; this
would give the perfectly comprehensible title, ' On a difference
in an angle, or on the contact of a circle and a sphere'. We
know from Aristotle that Protagoras, who wrote a book on
mathematics, ncpl tcov paOrj/idTow, used against the geometers
the argument that no such straight lines and circles as
they assume exist in nature, and that (e.g.). a material circle
does not in actual fact touch a ruler at one point only 2 ; and
it seems probable that Democritus's work was directed against
this sort of attack on geometry.
We know nothing of the contents of Democritus's book
On Geometry or of his Geometrica. One or other of these
works may possibly have contained the famous dilemma about
sections of a cone parallel to the base and very close together,
which Plutarch gives on the authority of Chrysippus. 3
' If, said Democritus, 'a cone were cut by a plane parallel
to the base [by which is clearly meant a plane indefinitely
1 Arist. Anal Pr. i. 24, 41 b 13-22.
2 Arist. Metaph. B. 2, 998 a 2.
3 Plutarch, De comm. not. adv. Stoicos, xxxix. 3.
N 2
180 THE ELEMENTS DOWN TO PLATO'S TIME
near to the base], what must we think of the surfaces forming
the sections? Are they equal or unequal? For, if they are
unequal, they will make the cone irregular as having many
indentations, like steps, and unevennesses ; but, if they are
equal, the sections will be equal, and the cone will appear to
have the property of the cylinder and to be made up of equal,
not unequal, circles, which is very absurd.'
The phrase ' made up of equal . . . circles ' shows that
Democritus already had the idea of a solid being the sum of
an infinite number of parallel planes, or indefinitely thin
laminae, indefinitely near together : a most important an-
ticipation of the same thought which led to such fruitful
results in Archimedes. This idea may be at the root of the
argument by which Democritus satisfied himself of the truth
of the two propositions attributed to him by Archimedes,
namely that a cone is one third part of the cylinder, and
a pyramid one third of the prism, which has the same base
and equal height. For it seems probable 1 that Democritus
would notice that, if two pyramids having the same height
and equal triangular bases are respectively cut by planes
parallel to the base and dividing the heights in the same
ratio, the corresponding sections of the two pyramids are
equal, whence he would infer that the pyramids are equal as
being the sum of the same infinite number of equal plane
sections or indefinitely thin laminae. (This would be a par-
ticular anticipation of Cavalieri's proposition that the areal or
solid content of two figures is equal if two sections of them
taken at the same height, whatever the height may be, always
give equal straight lines or equal surfaces respectively.) And
Democritus would of course see that the three pyramids into
which a prism on the same base and of equal height with the
original pyramid is divided (as in Eucl. XII. 7) satisfy this
test of equality, so that the pyramid would be one third part
of the prism. The extension to a pyramid with a polygonal
base would be easy. And Democritus may have stated the
proposition for the cone (of course without an absolute proof)
as a natural inference from the result of increasing indefinitely
the number of sides in a regular polygon forming the base of
a pyramid.
Tannery notes the interesting fact that the order in the list
4 DEM0CR1TUS 181
of Democritus's works of a the treatisesO n Geometry, Geometrica,
Numbers, and On irrational lines and solids corresponds to
the order of the separate sections of Euclid's Elements, Books
I- VI (plane geometry), Books VII-IX (on numbers), and
Book X (on irrationals). With regard to the work On irra-
tional lines and solids it is to be observed that, inasmuch as
his investigation of the cone had brought Deinocritus con-
sciously face to face with infinitesimals, there is nothing
surprising in his having written on irrationals ; on the con-
trary, the subject is one in which he would be likely to take
special interest. It is useless to speculate on what the treatise
actually contained ; but of one thing we may be sure, namely
that the dXoyoi ypa/x/xat, 'irrational lines', were not clto/jlol
ypafjLfxai, 'indivisible lines'. 1 Democritus was too good a
mathematician to have anything to do with such a theory.
We do not know what answer he gave to his puzzle about the
cone ; but his statement of the dilemma shows that he was
fully alive to the difficulties connected with the conception of
the continuous as illustrated by the particular case, and he
cannot have solved it, in a sense analogous to his physical
theory of atoms, by assuming indivisible lines, for this would
have involved the inference that the consecutive parallel
sections of the cone are unequal, in which case the surface
would (as he said) be discontinuous, forming steps, as it were.
Besides, we are told by Simplicius that, according to Demo-
critus himself, his atoms were, in a mathematical sense
divisible further and in fact ad infinitum, 2 while the scholia
to Aristotle's Be caelo implicitly deny to Democritus any
theory of indivisible lines : ' of those who have maintained
the existence of indivisibles, some, as for example Leucippus
and Democritus, believe in indivisible bodies, others, like
Xenocrates, in indivisible lines \ 3
With reference to the ^KTrerdorixaTa it is to be noted that
this word is explained in Ptolemy's Geography as the projec-
tion of the armillary sphere upon a plane. 4 This work and
that On irrational lines would hardly belong to elementary
geometry.
1 On this cf. 0. Apelt, Beitrcige zur Geschichte der griechischen Philo-
sophic, 1891, p. 265 sq.
2 Simpl. in Phys., p. 83. 5. 3 Scholia in Arist., p. 469 b 14, Brandis.
4 Ptolemy, Geogr. vii. 7.
182 THE ELEMENTS DOWN TO PLATO'S TIME
Hippias of Elis, the famous sophist already mentioned (pp. 2,
23-4), was nearly contemporary with Socrates and Prodicus,
and was probably born about 460 B.C. Chronologically, there-
fore, his place would be here, but the only particular discovery
attributed to him is that of the curve afterwards known as
the quadratrix, and the quadratrix does not come within the
-scope of the Elements. It was used first for trisecting any
rectilineal angle or, more generally, for dividing it in any
ratio whatever, and secondly for squaring the circle, or rather
for finding the length of any arc of a circle ; and these prob-
lems are not what the Greeks called ' plane ' problems, i. e.
they cannot be solved by means of the ruler and compasses.
It is true that some have denied that the Hippias who
invented the quadratrix can have been Hippias of Elis ;
Blass 1 and Apelt 2 were of this opinion, Apelt arguing that at
the time of Hippias geometry had not got far beyond the
theorem of Pythagoras. To show how wide of the mark this
last statement is we have only to think of the achievements
of Democritus. We know, too, that Hippias the sophist
specialized in mathematics, and I agree with Cantor and
Tannery that there is no reason to doubt that it was he who
discovered the quadratrix. This curve will be best described
when we come to deal with the problem of squaring the circle
(Chapter VII) ; here we need only remark that it implies the
proposition that the lengths of arcs in a circle are proportional
to the angles subtended by them at the centre (Eucl. VI. 33).
The most important name from the point of view of this
chapter' is Hippocrates of Chios. He is indeed the first
person of whom it is recorded that he compiled a book of
Elements. This is lost, but Simplicius has preserved in his
commentary on the Physics of Aristotle a fragment from
Eudemus's History of Geometry giving an account of Hippo-
crates's quadratures of certain ' lunules ' or lunes. 3 This is one
of the most precious sources for the history of Greek geometry
before Euclid ; and, as the methods, with one slight apparent
exception, are those of the straight line and circle, we can
form a good idea of the progress which had been made in the
Elements up to Hippocrates's time.
1 Fleckeisen's Jahrbuch, cv, p. 28.
2 Beitrcige zuv Gesch. d. gr. Philoso2)hie, p. 379.
3 Simpl. in Phi/s., pp. 60. 22-68. 32, Diels.
HIPPOCRATES OF CHIOS 183
It would appear that Hippocrates was in Athens during
a considerable portion of the second half of the fifth century,
perhaps from 450 to 430 B.C. We have quoted the story that
what brought him there was a suit to recover a large sum
which he had lost, in the course of his trading operations,
through falling in with pirates; he is said to have remained
in Athens on this account a long time, during which he con-
sorted with the philosophers and reached such a degree of
proficiency in geometry that he tried to discover a method of
squaring the circle. 1 This is of course an allusion to the
quadratures of lunes.
Another important discovery is attributed to Hippocrates.
He was the first to observe that the problem of doubling the
cube is reducible to that of finding two mean proportionals in
continued proportion between two straight lines. 2 The effect
of this was, as Proclus says, that thenceforward people
addressed themselves (exclusively) to the equivalent problem
of finding two mean proportionals between two straight lines. 3
(a) Hippocrates s quadrature of lunes.
I will now give the details of the extract from Eudemus on
the subject of Hippocrates's quadrature of lunes, which (as
I have indicated) I place in this chapter because it belongs
to elementary ' plane ' geometry. Simplicius says he will
quote Eudemus ' word for word ' (Kara X^lu) except for a few
additions taken from Euclid's Elements, which he will insert
for clearness' sake, and which are indeed necessitated by the
summary (memorandum-like) style of Eudemus, whose form
of statement is condensed, 'in accordance with ancient prac-
tice'. We have therefore in the first place to distinguish
between what is textually quoted from Eudemus and what
Simplicius has added. To Bretschneider 4 belongs the credit of
having called attention to the importance of the passage of
Simplicius to the historian of mathematics ; Allman 5 was the
first to attempt the task of distinguishing between the actual
1 Philop. in Phys., p. 31. 3, Vitelli.
2 Pseudo-Eratosthenes to King Ptolemy in Eutoc. on Archimedes (vol.
iii, p. 88, Heib.).
3 Proclus on Eucl. I, p. 213. 5.
4 Bretschneider, Die Geometrie and die Geometer vor EuJdides, 1870,
pp. 100-21.
5 Hermathena, iv, pp. 180-228; Greek Geometry from Thales to Euclid,
pp. 64-75.
184 THE ELEMENTS DOWN TO PLATO'S TIME
extracts from Eudemus and Simplicius's amplifications ; then
came the critical text of Simplicius's commentary on the
Physics edited by Diels (1882), who, with the help of Usener,
separated out, and marked by spacing, the portions whiclf they
regarded as Eudemus's own. Tannery, 1 who had contributed
to the preface of Diels some critical observations, edited
(in 1883), with a translation and notes, what he judged to be
Eudemian (omitting the rest). Heiberg' 2 reviewed the whole
question in 1884; and finally Rudio, 3 after giving in the
Blbliotheca Mathematica of 1902 a translation of the whole
passage of Simplicius with elaborate notes, which again he
followed up by other articles in the same journal and elsewhere
in 1903 and 1905, has edited the Greek text, with a transla-
tion, introduction, notes, and appendices, and summed up the
whole controversy.
The occasion of the whole disquisition in Simplicius's com-
mentary is a remark by Aristotle that there is no obligation
on the part of the exponent of a particular subject to refute
a fallacy connected with it unless the author of the fallacy
has based his argument on the admitted principles lying at
the root of the subject in question. ' Thus ', he says, ' it is for
the geometer to refute the (supposed) quadrature of a circle by
means of segments (r/x^/zaro)^), but it is not the business of the
geometer to refute the argument of Antiphon.' 4 Alexander
took the remark to refer to Hippocrates's attempted quadra-
ture by means of lunes (although in that case T/irj/xa is used
by Aristotle,' not in the technical sense of a segment, but with
the non-technical meaning of any portion cut out of a figure).
This, probable enough in itself (for in another place Aristotle
uses the same word T/irj/xa to denote a sector of a circle 5 ), is
made practically certain by two other allusions in Aristotle,
one to a proof that a circle together with certain lunes is
equal to a rectilineal figure, 6 and the other to ' the (fallacy) of
Hippocrates or the quadrature by means of the lunes '. 7 The
1 Tannery, Memoires scientifiques, vol. i, 1912, pp. 339-70, esp. pp.
347-66.
2 Philologus, 43, pp. 336-44.
3 Rudio, Der Bericht des Simplicius ilber die Quadraturen des Antiphon
und Hippokrates (Teubner, 1907).
4 Arist. Phys. i. 2, 185 a 14-17. 5 Arist. De caelo, ii. 8, 290 a 4.
6 Anal. Pr. ii. 25, 69 a 32. 7 Soph. til. 11, 171 b 15.
HIPPOCRATES'S QUADRATURE OF LUNES 185
two expressions separated by ' or ' may no doubt refer not to
one but to two different fallacies. But if ' the quadrature by
means of lunes ' is different from Hippocrates's quadratures of
lunes, it must apparently be some quadrature like the second
quoted by Alexander (not by Eudemus), and the fallacy attri-
buted to Hippocrates must be the quadrature of a certain lune
plus a circle (which in itself contains no fallacy at all). It seems
more likely that the two expressions refer to one thing, and that
this is the argument of Hippocrates's tract taken as a whole.
The passage of Alexander which Simplicius reproduces
before passing to the extract from Eudemus contains two
simple cases of quadrature, of a lune, and of lunes plus a semi-
circle respectively, with an erroneous inference from these
cases that a circle is thereby squared. It is evident that this
account does not represent Hippocrates's own argument, for he
would not have been capable of committing so obvious an
error ; Alexander must have drawn his information, not from
Eudemus, but from some other source. Simplicius recognizes
this, for, after giving the alternative account extracted from
Eudemus, he says that we must trust Eudemus's account rather
than the other, since Eudemus was 'nearer the times' (of
Hippocrates).
The two quadratures given by Alexander are as follows.
1. Suppose that AB is the diameter of a circle, D its centre,
and AC, CB sides of a square
inscribed in it.
On iC as diameter describe
the semicircle A EC. Join CD.
Now, since
AB 2 = 2 AC 2 ,
A O B
and circles (and therefore semi-
circles) are to one another as the squares on their diameters,
(semicircle ACB) = 2 (semicircle AEC).
But (semicircle ACB) = 2 (quadrant ADC) ;
therefore (semicircle A EC)= (quadrant ADC).
If now we subtract the common part, the segment AFC,
we have (lune AECF) = AADC,
and the lune is ' squared '.
186 THE ELEMENTS DOWN TO PLATO'S TIME
2. Next take three consecutive sides CE, EF, FD of a regular
hexagon inscribed in a circle of diameter CD. Also take A B
equal to the radius of the circle and therefore equal to each of
the sides.
On AB, GE, EF, FD as diameters describe semicircles (in
the last three cases outwards with reference to the circle).
Then, since
CD 2 = 4 AB 2 = A B 2 + CE 2 + EF 2 + FD 2 ,
and circles are to one another as the squares on their
diameters,
semicircle GEFD) = 4 (semicircle ALB)
= (sum of semicircles ALB, GGE, EHF, FKD).
H
Subtracting from each side the sum of the small segments
on CE, EF, FD, we have
(trapezium GEFD) = (sum of three lunes) -f (semicircle ALB).
The author goes on to say that, subtracting the rectilineal
figure equal to the three lunes (' for a rectilineal figure was
proved equal to a lune '), we get a rectilineal figure equal
to the semicircle ALB, 'and so the circle will have been
squared '.
This 'conclusion is obviously false, and, as Alexander says,
the fallacy is in taking what was proved only of the lune on
the side of the inscribed square, namely that it can be squared,
to be true of the lunes on the sides of an inscribed regular
hexagon. It is impossible that Hippocrates (one of the ablest
of geometers) could have made such a blunder. We turn there-
fore to Eudemus's account, which has every appearance of
beginning at the beginning of Hippocrates's work and pro-
ceeding in his order.
HIPPOCRATES'S QUADRATURE OF LUNES 187
It is important from the point of view of this chapter to
preserve the phraseology of Eudemus, which throws light
on the question how far the technical terms of Euclidean
geometry were already used by Eudemus (if not by Hippo-
crates) in their technical sense. I shall therefore translate
literally so much as can safely be attributed to Eudemus
himself, except in purely geometrical work, where I shall use
modern symbols.
' The quadratures of lunes, which were considered to belong
to an uncommon class of propositions on account of the
close relation (of lunes) to the circle, were first investigated
by Hippocrates, and his exposition was thought to be in
correct form 1 ; we will therefore deal with them at length
and describe them. He started with, and laid down as the
first of the theorems useful for his purpose, the proposition
that similar segments of circles have the same ratio to one
another as the squares on their bases have [lit. as their bases
in square, 6vvd/i€i\. And this he proved by first showing
that the squares on the diameters have the same ratio as the
circles. TFor, as the circles are to one another, so also are
similar segments of them. For similar segments are those
which are the same part of the circles respectively, as for
instance a semicircle is similar to a semicircle, and a third
part of a circle to a third part [here, Rudio argues, the word
segments, Tfirj^aTa, would seem to be used in the sense of
sector's]. It is for this reason also (Sib kol) that similar
segments contain equal angles [here ' segments ' are certainly
segments in the usual sense]. The angles of all semicircles
are right, those of segments greater than a semicircle are less
than right angles and are less in proportion as the segments
are greater than semicircles, while those of segments less than
a semicircle are greater than right angles and are greater in
proportion as the segments are less than semicircles.']
I have put the last sentences of this" quotation in dotted
brackets because it is matter of controversy whether they
belong to the original extract from Eudemus or were added by
Simplicius.
I think I shall bring out the issues arising out of this
passage into the clearest relief if I take as my starting-point
the interpretation of it by Rudio, the editor of the latest
1 Kara rponov ( ' werthvolle Abhandlung \ Heib.).
188 THE ELEMENTS DOWN TO PLATO'S TIME
edition of the whole extract. Whereas Diels, Usener, Tannery,
and Heiberg had all seen in the sentences ' For, as the circles
are to one another . . . less than semicircles' an addition by
Simplicius, like the phrase just preceding (not quoted above),
' a proposition which Euclid placed second in his twelfth book
with the enunciation " Circles are to one another as the squares
on their diameters " ', Rudio maintains that the sentences are'
wholly Eudemian, because ' For, as the circles are to one
another, so are the similar segments' is obviously connected
with the proposition that similar segments are as the squares
on their bases a few lines back.- Assuming, then, that the
sentences are Eudemian, Rudio bases his next argument on,
the sentence defining similar segments, ' For similar segments
are those which are the same part of the circles : thus a semi-
circle is similar to a semicircle, and a third part (of one circle)
to a third part (of another circle) '. He argues that a ' segment '
in the proper sense which is one third, one fourth, &c, of the
circle is not a conception likely to have been introduced into
Hippocrates's discussion, because it cannot be visualized by
actual construction, and so would not have conveyed any clear
idea. On the other hand, if we divide the four right angles
about the centre of a circle into 3, 4, or n equal parts by
means of 3, 4, or n radii, we have an obvious division of the
circle into equal parts which would occur to any one ; that is,
any one would understand the expression one third or one
fourth part of a circle if the parts were sectors and not
segments. (The use of the word Tfifjfia in the sense of sector
is not impossible in itself at a date when mathematical
terminology was not finally fixed ; indeed it means ' sector '
in one passage* of Aristotle. 1 ) Hence Rudio will have it that
'similar segments' in the second and third places in our passage
are ' similar sectors '. * But the ' similar segments ' in the funda-
mental proposition of Hippocrates enunciated just before are
certainly segments in the proper sense : so are those in the
next sentence which says that similar segments contain equal
angles. There is, therefore, the very great difficulty that,
under Rudio's interpretation, the word T/xrjfj.aTa used in
successive sentences means, first segments, then sectors, and
then segments again. However, assuming this to be so, Rudio
1 Arist. De caelo, ii. 8, 290 a 4.
HIPPOCRATES'S QUADRATURE OF LUNES 189
is able to make the argument hang together, in the following
way. The next sentence says, ' For this reason also (8lo kgu)
similar segments contain equal angles ' ; therefore this must be
inferred from the fact that similar sectors are the same part
of the respective circles. The intermediate steps are not given
in the text; but, since the similar sectors are the same part
of the circles, they contain equal angles, and it follows that the
angles in the segments which form part of the sectors are
equal, since they are the supplements of the halves of the
angles of the sectors respectively (this inference presupposes
that Hippocrates knew the theorems of Eucl. III. 20-22, which
is indeed clear from other passages in the Eudemus extract).
Assuming this to be the line of argument, Rudio infers that in
Hippocrates's time similar segments were not defined as in
Euclid (namely as segments containing equal angles) but were
regarded as the segments belonging to ' similar sectors ', which
would thus be the prior conception. Similar sectors would
be sectors having their angles equal. The sequence of ideas,
then, leading up to Hippocrates's proposition would be this.
Circles are to one another as the squares on their diameters or
radii. Similar sectors, having their angles equal, are to one
another as the whole circles to which they belong. (Euclid has
not this proposition, but it is included in Theon's addition to
VI. 33, and would be known long before Euclid's time.)
Hence similar sectors are as the squares on the radii. But
so are the triangles formed by joining the extremities of the
bounding radii in each sector. Therefore (cf. Eucl. V. 19)
the differences between the sectors and the corresponding-
triangles respectively, i.e. the corresponding segments, are in
the same ratio as (1) the similar sectors, or (2) the similar
triangles, and therefore are as the squares on the radii.
We could no doubt accept this version subject to three ifs,
(1) if the passage is Eudemian, (2) if we could suppose
T\ir\\iara to be used in different senses in consecutive sentences
without a word of explanation, (3) if the omission of the step
between the definition of similar ' segments ' and the inference
that the angles in similar segments are equal could be put
down to Eudemus's ' summary ' style. The second of these
ifs is the crucial one ; and, after full reflection, I feel bound
to agree with the great scholars who have held that this
190 THE ELEMENTS DOWN TO PLATO'S TIME
hypothesis is impossible ; indeed the canons of literary criti-
cism seem to exclude it altogether. If this is so, the whole
of Rudio's elaborate structure falls to the ground.
We can now consider the whole question ah initio. First,
are the sentences in question the words of Eudemus or of
Simplicius ? On the one hand, I think the whole paragraph
would be much more like the ' summary ' manner of Eudemus
if it stopped at 'have the same ratio as the circles', i.e. if the
sentences were not there at all. Taken together, they are
long and yet obscurely argued, while the last sentence is
really otiose, and, I should have said, quite unworthy of
Eudemus. On the other hand, I do not see that Simplicius
had any sufficient motive for interpolating such an explana-
tion : he might have added the words ' for, as the circles are
to one another, so also are similar segments of them ', but
there was no need for him to define similar segments ; he
must have been familiar enough with the term and its
meaning to take it for granted that his readers would know
them too. I think, therefore, that the sentences, down to ' the
same part of the circles respectively ' at any rate, may be
from Eudemus. In these sentences, then, can ' segments ' mean
segments in the proper sense (and not sectors) after all ?
The argument that it cannot rests on the assumption that the
Greeks of Hippocrates's day would not be likely to speak of
a segment which was one third of the whole circle if they
did not see their way to visualize it by actual construction.
But, though the idea would be of no use to us, it does not
follow that their point of view would be the same as ours.
On the contrary, I agree with Zeuthen that Hippocrates may
well have said, of segments of circles which are in the same
ratio as the circles, that they are ' the same part ' of the circles
respectively, for this is (in an incomplete form, it is true) the
language of the definition of proportion in the only theory of
proportion (the numerical) then known (cf. Eucl. VII. Def. 20,
' Numbers are proportional when the first is the same multiple,
or the same part, or the same parts, of the second that the
third is of the fourth', i.e. the two equal ratios are of one
T)\i
of the following forms m, or — where m, n are integers) ;
the illustrations, namely the semicircles and the segments
HIPPOCRATES'S QUADRATURE OF LUNES 191
which are one third of the circles respectively, are from this
point of view quite harmless.
Only the transition to the view of similar segments as
segments ' containing equal angles ' remains to be explained.
And here we are in the dark, because we do not know how, for
instance, Hippocrates would have drawn a segment in one
given circle which should be ' the same part ' of that circle
that a given segment of another given circle is of that circle.
(If e.g. he had used the proportionality of the parts into which
the bases of the two similar segments divide the diameters
of the circles which bisect them perpendicularly, he could,
b}^ means of the sectors to which the segments belong, have
proved that the segments, like the sectors, are in the ratio
of the circles, just as Rudio supposes him to have done ; and
the equality of the angles in the segments would have followed
as in Rudio's proof.)
As it is, I cannot feel certain that the sentence Slo kcu ktX.
' this is the reason why similar segments contain equal angles '
is not an addition by Simplicius. Although Hippocrates was
fully aware of the fact, he need not have stated it in this
place, and Simplicius may have inserted the sentence in order
to bring Hippocrates's view of similar segments into relation
with Euclid's definition. TJie sentence which follows about
'angles of semicircles and 'angles of segments, greater or
less than semicircles, is out of place, to say the least, and can
hardly come from Eudemus.
We resume Eudemus's account.
' After proving this, he proceeded to show in what way it
was possible to square a lune the outer circumference of which
is that of a semicircle. This he effected by circumscribing
a semicircle about an isosceles right-angled triangle and
(circumscribing) about the base [ = describing on the base 1
a segment of a circle similar to those cut off by the sides.'
[This is the problem of Eucl. III. 33,
and involves the knowledge that similar
segments contain equal angles.]
'Then, since the segment about the
base is equal to the sum of those about
the sides, it follows that, when the part
of the triangle above the segment about the base is added
to both alike, the lune will be equal to the triangle.
192 THE ELEMENTS DOWN TO PLATO'S TIME
' Therefore the lime, having been proved, equal to the triangle,
can be squared.
' In this way, assuming that the outer circumference of
the lune is that of a semicircle, Hippocrates easily squared
the lune.
' Next after this he assumes (an outer circumference) greater
than a semicircle (obtained) by constructing a trapezium in
which three sides are equal to one another, while one, the
greater of the parallel sides, is such that the square on it is
triple of the square on each one of the other sides, and then
comprehending the trapezium in a circle and circumscribing
about (= describing on) its greatest side a segment similar
to those cut off from the circle by
the three equal sides.' *
[Simplicius here inserts an easy
proof that a circle can be circum-
scribed about the trapezium. 1 ]
' That the said segment [bounded
by the outer circumference BACD
in the figure] is greater than a
semicircle is clear, if a diagonal
be drawn in the trapezium.
' For this diagonal [say BC\
subtending two sides [BA, AC] of
the trapezium, is such that the
square on it is greater than double
the square on one of the remain-
ing sides.'
[This follows from the fact that, AC being parallel to
BD but less than it, BA and DC will meet, if produced, fh
a point F. Then, in the isosceles triangle FAC, the angle
FAC is less than a right angle, so that the angle BAC is
obtuse.]
' Therefore the square on [BD] the greatest side of the trape-
zium [= 3 CD 2 by hypothesis] is less than the sum of the
squares on the diagonal [BC] and that one of the other sides
1 Heiberg {Philologus, 43, p. 340) thinks that the words kcu on pev
7repi\r)(f>dr)<TeTa.i kvkXcoto Tpcnre(iov &(ii~eis [ovrcot] dt^ompqaas to? tov Tpmrc(lov
ycoiias ('Now, that the trapezium can be comprehended in a circle you
can prove by bisecting the angles of the trapezium ') may (without ovtcos—
F omits it) be Eudemus's own . For on peu . . . forms a natural contrast
to on 8i fulCov ... in the next paragraph. Also cf. p. 65. 9 Diels, tovtcov
ovu ovtoos €\ovtu)V to Tjjmre(i6v (prjfxi tcp* ov EKBH ■ncpikrjtyeTai kvkXos.
HIPPOCRATES'S QUADRATURE OF LUNES 193
[CD] which is subtended 1 by the said (greatest) side [BD]
together with the diagonal [BC] ' [i.e. BD 2 < BC 2 + CD 2 ].
'Therefore the angle standing on the greater side of the
trapezium [Z BCD] is acute.
' Therefore the segment in which the said angle is is greater
than a semicircle. And this (segment) is the outer circum-
ference of the lune/
[Simplicius observes that Eudemus has omitted the actual
squaring of the lune, presumably as being obvious. We have
only to supply the following.
Since BD 2 = 3 BA 2 ,
(segment on BD) = 3 (segment on BA)
= (sum of segments on BA, AC, CD).
Add to each side the area between BA, AC, CD, and the
circumference of the segment on BD, and we have
(trapezium A BDC) = (lune bounded by the two circumferences).]
' A case too where the outer circumference is less than
a semicircle was solved by Hippocrates, 2 who gave the follow-
ing preliminary construction.
' Let there be a circle %vith diameter AB, and let its centre
be K.
{ Let CD bisect BK at right angles; and let the straight
line EF be so placed between CD and the circumference that it
verges towards B [i.e. will, if produced, pass through B], while
its length is also such that the square on it is 1~ times the square
on (one of) the radii.
1 Observe the curious use of imoTeiveiv, stretch under, subtend. The
third side of a triangle is said to be ' subtended ' by the other two
together.
2 Literally ' If (the outer circumference) were less than a semicircle,
Hippocrates solved (KaTfo-Kevaaev, constructed) this (case).'
1523 O
194 THE ELEMENTS DOWN TO PLATO'S TIME
( Let EG be drawn parallel to AB, and let (straight lines)
be dravm joining K to E and F.
' Let the straight line \KF~\ joined to F and produced meet
EG in G, and again let (straight lines) be drawn joining
B to F, G.
1 It is then manifest that BF produced will pass through
[" fall on "] E [for by hypothesis EF verges towards B], and
BG will be equal to EK.'
[Simplicius proves this at length. The proof is easy. The
triangles FKC, FBG are equal in all respects [Eucl. I. 4].
Therefore, EG being parallel to KB, the triangles EDF, GDF
are equal in all respects [Eucl. I. 15, 29, 26]. Hence the
trapezium is isosceles, and BG = EK.
' This being so, I say that the trapezium EKBG can be
comprehended in a circle!
[Let the segment EKBG circumscribe it.]
' Next let a segment of a circle be circumscribed about the
triangle EFG also ;
then manifestly each of the segments [on] EF, FG will be
similar to each of the segments [on] EK, KB, BG.'
[This is because all the segments contain equal angles,
namely an angle equal to the supplement of EGK .]
' This being so, the lune so formed, of which EKBG is the
outer circumference, will be equal to the rectilineal figure made
up of the three triangles BFG, BFK, EKF.
' For the segments cut off from the rectilineal figure, on the
inner side of the lune, by the straight lines EF, FG, are
(together) equal to the segments outside the rectilineal figure
cut off by the straight lines EK, KB, BG, since each of the
inner segments is l-§ times each of the outer, because, by
hypothesis, EF 2 ( _ FG 2j = * E K2
[i.e. 2EF 2 = 3EK*,
= EK 2 + KB 2 + BG 2 ].
< If then
(lune) = (the three segmts.) + {(rect. fig.) — (the two segmts.) },
the trapezium including the two segments but not the three,
while the (sum of the) two segments is equal to the (sum
of the) three, it follows that
(lune) = (rectilineal figure).
HIPPOCRATES'S QUADRATURE OF LUNES 195
'The fact that this lune (is one which) has its outer circum-
ference less than a semicircle he proves by means of the fact
that the angle [EK 6r] in the outer segment is obtuse.
' And the fact that the angle EKG is obtuse he proves as
follows.'
[This proof is supposed to have been given by Eudemus in
Hippocrates's own words, but unfortunately the text is con-
fused. The argument seems to have been substantially as
follows.
By hypothesis, EF 2 = § EK 2 .
Also BK 2 > 2BF 2 (this is assumed: we shall
consider the ground later) ;
or EK 2 > 2KF 2 .
Therefore EF 2 = EK 2 + J EK 2
> EK 2 + KF\
so that the angle EKF is obtuse, and the segment is less than
a semicircle.
How did Hippocrates prove that BK 2 > 2 BF 2 1 The manu-
scripts have the phrase ' because the angle at F is greater' (where
presumably we should supply opOrj?, 'than a right angle').
But, if Hippocrates proved this, he must evidently have proved
it by means of his hypothesis EF 2 = \EK 2 , and this hypo-
thesis leads more directly to the consequence that- BK 2 > 2KF 2
than to the fact that the angle at F is greater than a right
angle.
We may supply the proof thus.
By hypothesis, EF 2 = § KB 2 .
Also, since A, E, F, G are concyclic,
EB.BF= AB.BG
= KB 2 ,
or EF.FB + BF 2 = KB 2
= § EF 2 .
It follows from the last relations that EF > FB, and that
KB 2 >2BF 2 .
The most remarkable feature in the above proof is the
assumption of the solution of the problem ' to place a straight
o2
196 THE ELEMENTS DOWN TO PLATO'S TIME
line [EF] of length such that the square on it is l-§ times the
square on AK between the circumference of the semicircle and
CD in such a way that it will verge (iteveiv) towards B ' [i.e. if
produced, will pass through B\ This is a problem of a type
which the Greeks called vevaeis, inclinationes or vergings.
Theoretically it may be regarded as the problem of finding
a length (x) such that, if F be so taken on CD that BF = x,
BF produced will intercept between CD and the circumference
of the semicircle a length EF equal to \/§ . AK.
If we suppose it done, we have
EB.BF=AB.BC = AK 2 ;
or x (x + \/f . a) = a 2 (where AK = a).
That is, the problem is equivalent to the solution of the
quadratic equation
x 2 + \/-§ . ax = a 2 .
This again is the problem of 'applying to a straight line
of length </§ , a a rectangle exceeding by a square figure and
equal in area to a 2 ', and would theoretically be solved by the
Pythagorean method based on the theorem of Eucl. II. 6.
Undoubtedly Hippocrates could have solved the problem by
this theoretical method ; but he may, on this occasion, have
used the purely mechanical method of marking on a ruler
or straight edge a length equal to \/f . AK, and then moving
it till the points marked lay on the circumference" and on CD
respectively, while the straight edge also passed through B.
This method is perhaps indicated by the fact that he first
'places EF (without producing it to B) and afterwards
joins BF.
We come now to the last of Hippocrates's quadratures.
Eudemus proceeds:]
' Thus Hippocrates squared every l (sort of) lune, seeing
that 1 (he squared) not only (1) the lune which has for its outer
1 Tannery brackets iravrn and eiWp kgl. Heiberg thinks (l.c , p. 343)
the wording is that of Simplicius reproducing the content of Eudemus.
The wording of the sentence is important with reference to the questions
(1) What was the paralogism with which Aristotle actually charged
Hippocrates ? and (2) What, if any, was the justification for the charge ?
Now the four quadratures as given by Eudemus are clever, and contain in
themselves no fallacy at all. The supposed fallacy, then, can only have
consisted in an assumption on the part of Hippocrates that, because he
HIPPOCRATES'S QUADRATURE OF LUNES 19?
circumference the arc of a semicircle, but also (2) the lune
in which the outer circumference is greater, and (3) the lune in
which it is less, than a semicircle.
' But he also squared the sum of a lune and a circle in the
following manner.
' Let there be Mvo circles about K as centime, such that the
square on the diameter of the outer is 6 times the square on
that of the inner.
'Let a (regular) hexagon ABCDEF be inscribed in the
inner circle, and let KA, KB, KG be joined from the centre
and produced as far as the circumference of the outer circle.
Let GH, HI, GI be joined!
[Then clearly GH, HI are sides of a hexagon inscribed in
the outer circle.]
'About GI [i.e. on GI] let a segment be circumscribed
similar to the segment cut off by GH.
'Then GI 2 =SGH 2 ,
for GI 2 + (side of outer hexagon) 2 = (diam. of outer circle) 2
= 4GH 2 .
[The original states this in words without the help of the
letters of the figure.]
'Also GH 2 = QAB 2 .
had squared one particular lune of each of three types, namely those
which have for their outer circumferences respectively (1) a semicircle,
(2) an arc greater than a semicircle, (3) an arc less than a semicircle, he
had squared all possible lunes, and therefore also the lune included in his
last quadrature, the squaring of which (had it been possible) would
actually have enabled him to square the circle. The question is, did
Hippocrates so delude himself? Jffeiberg thinks that, in the then
state of logic, he may have done so. But it seems impossible to believe
this of so good a mathematician ; moreover, if Hippocrates had really
thought that he had squared the circle, it is inconceivable that he
would not have said so in express terms at the end of his fourth
quadrature.
Another recent view is that of Bjornbo (in Pauly-Wissowa, Keal-Ency-
clopadie, xvi, pp. 1787-99), who holds that Hippocrates realized perfectly
the limits of what he had been able to do and knew that he had not
squared the circle, but that he deliberately used language which, without
being actually untrue, was calculated to mislead any one who read him
into the belief that he had really solved the problem. This, too, seems
incredible ; for surely Hippocrates must have known that the first expert
who read his tract would detect the fallacy at once, and that he was
risking his reputation as a mathematician for no purpose. I prefer to
think that he was merely trying to put what he had discovered in the
most favourable light ; but it must be admitted that the effect of his
language was only to bring upon himself a charge which he might easily
have avoided.
198 THE ELEMENTS DOWN TO PLATO'S TIME
' Therefore
segment on GI[ = 2(segmt. on GH) + 6 (segmt. on AB)\
= (segmts. on GH, HI) + (all segmts. in
inner circle).
[' Add to each side the area bounded by GH, HI and the
arc GI;]
therefore (A GHI) = (lune GHI) + (all segmts. in inner circle).
Adding to both sides the hexagon in the inner circle, we have
(A GHI) + (inner hexagon) = (lune GHI) + (inner circle).
' Since, then, the sum of the two rectilineal figures can be
squared, so can the sum of the circle and the lune in question.'
Simplicius adds the following observations :
' Now, so far as Hippocrates is concerned, we must allow
that Eudemus was in a better position to know the facts, since
he was nearer the times, being a pupil of Aristotle. But, as
regards the " squaring of the circle by means of segments "
which Aristotle reflected on as containing a fallacy, there are
three possibilities, (1) that it indicates the squaring by means
of lunes (Alexander was quite right in expressing the doubt
implied by his words, "if it is the same as the squaring by
means of lunes"), (2) that it refers, not to the proofs of
Hippocrates, but some others, one of which Alexander actually
reproduced, or (3) that it is intended to reflect on the squaring
by Hippocrates of the circle plus the lune, which Hippocrates
did in fact prove "by means of segments", namely the three
(in the greater circle) and those in the lesser circle. . . . On
HIPPOCRATES'S QUADRATURE OF LUNES 199
this third hypothesis the fallacy would lie in the fact that
the sum of the circle and the lune is squared, and not the
circle alone.'
If, however, the reference of Aristotle was really to Hip-
pocrates's last quadrature alone, Hippocrates was obviously
misjudged ; there is no fallacy in it, nor is Hippocrates likely
to have deceived himself as to what his proof actually
amounted to.
In the above reproduction of the extract from Eudemus
I have marked by italics the passages where the writer follows
the ancient fashion of describing points, lines, angles, &c, with
reference to the letters in the figure : the ancient practice was
to write to o-qfidov k(j> a> (or €</>' ov) K, the (point) on which (is)
the letter K, instead of the shorter form to K a-q^elov, the
point K, used by Euclid and later geometers; rj e0' 77 AB
(evOeia), the straight line on which (are the letters AB, for
rj AB (evOeia), the straight line AB; to Tpiyoavov to e0' ov
EZH, the triangle on which (are the letters) EFG, instead of
to EZH Tplycavov, the triangle EFG ; and so on. Some have
assumed that, where the longer archaic form, instead of the
shorter Euclidean,is used, Eudemus must be quoting Hippocrates
verbatim ; but this is not a safe criterion, because, e.g., Aristotle
himself uses both forms of expression, and there are, on the
other hand, some relics of the archaic form even in Archimedes.
Trigonometry enables us readily to find all the types of
Hippocratean lunes that can
be squared by means of the
straight line and circle. Let
ACB be the external circum-
ference, ABB the internal cir- -
cumference of such a lune,
r, r' the radii, and 0, 0' the
centres of the two arcs, 0, 6'
the halves of the angles sub-
tended by the arcs at the centres
respectively.
Now (area of lune)
= (difference of segments ACB, ADB)
= (sector 0AGB- /\A0B)~ (sector O'ADB-AAO'B)
= T 20-r' 2 6' + i (r' 2 sin 2 6' - r 2 sin 2 0).
200 THE ELEMENTS DOWN TO PLATO'S TIME
We also have
rsinO = J-4J? = r'sin 6' . ..... (1)
In order that the lune may be squareable, we must have, in
the first place, r 2 = r' 2 d'.
Suppose that = m6', and it follows that
r' = Vm . r.
Accordingly the area becomes
■| r 2 (m sin 2 0' — sin 2m6 / ) ;
and it remains only to solve the equation (1) above, which
becomes Binm0'= */m.sin0'.
This reduces to a quadratic equation only when m has one
of the Values o ^ 5 5 -§.
The solutions of Hippocrates correspond to the first three
values of m. But the lune is squareable by ' plane ' methods
in the other two cases also. Clausen (1840) gave the last four
cases of the problem as new x (it was not then known* that
Hippocrates had solved more than the first) ; but, according
to M. Simon 2 , all five cases were given much earlier in
a dissertation by Martin Johan Wallenius of Abo (Abveae,
1766). As early as 1687 Tschirnhausen noted the existence
of an infinite number of squareable portions of the first of
Hippocrates's lunes. Vieta 3 discussed the case in which m = 4,
which of course leads to a cubic equation.
(fi) Reduction of the problem of doubling the cube to
the finding of two mean proportionals.
We have already alluded to Hippocrates's discovery of the
reduction of the problem of duplicating the cube to that of
finding two mean proportionals in continued proportion. That
is, he discovered that, if
a:x = x:y = y:b,
then a 3 : x 7, — a : b. This shows that he could work with
compound ratios, although for him the theory of proportion
must still have been the incomplete, numerical, theory
developed by the Pythagoreans. It has been suggested that
1 Crelle, xxi, 1840, pp. 375-6.
2 GeschicMe der Math, im Altertum, p. 174.
8 Vieta, Variorum de rebus mathematicis responsorum lib. viii, 1593.
ELEMENTS AS KNOWN TO HIPPOCRATES 201
the idea of the reduction of the problem of duplication may
have occurred to him through analogy. The problem of
doubling a square is included in that of finding one mean
proportional between two lines; he might therefore have
thought of what would be the effect of finding two mean
proportionals. Alternatively he may have got the idea from
the theory of numbers. Plato in the Timaeus has the pro-
positions that between two square numbers there is one mean
proportional number, but that two cube numbers are connected,
not by one, but by two mean numbers in continued proportion. 1
These are the theorems of Eucl. VIII. 11, 12, the latter of
which is thus enunciated : ' Between two cube numbers there
are two mean proportional numbers, and the cube has to the
cube the ratio triplicate of that which the side has to the side.'
If this proposition was really Pythagorean, as seems prob-
able enough, Hippocrates had only to give the geometrical
adaptation of it.
(y) The Elements as known to Hippocrates.
We can now take stock of the advances made in the
Elements up to the time when Hippocrates compiled a work
under that title. We have seen that the Pythagorean geometry
already contained the substance of Euclid's Books I and II,
part of Book IV, and theorems corresponding to a great part
of Book VI ; but there is no evidence that the Pythagoreans
paid much attention to the geometry of the circle as we find
it, e.g., in Eucl., Book III. But, by the time of Hippocrates,
the main propositions of Book III were also known and used,
as we see from Eudemus's account of the quadratures of
lunes. Thus it is assumed that ' similar ' segments contain
equal angles, and, as Hippocrates assumes that two segments
of circles are similar when the obvious thing about the figure
is that the angles at the circumferences which are the supple-
ments of the angles in the segments are one and the same,
we may clearly infer, as above stated, that Hippocrates knew
the theorems of Eucl. III. 20-2. Further, he assumes the
construction on a given straight line of a segment similar to
another given segment (cf. Eucl. III. 33). The theorems of
Eucl. III. 26-9 would obviously be known to Hippocrates
1 Plato, Timaeus, 32 A, b.
202 THE ELEMENTS DOWN TO PLATO'S TIME
as was that of III. 31 (that the angle in a semicircle is
a right angle, and that, according as a segment is less or
greater than a semicircle, the angle in it is obtuse or acute).
He assumes the solution of the problem of circumscribing
a circle about a triangle (Eucl. IV. 5), and the theorem that
the side of a regular hexagon inscribed in a circle is equal
to the radius (Eucl. IV. 15).
But the most remarkable fact of all is that, according to
Eudemus, Hippocrates actually proved the theorem of Eucl.
XII. 2, that circles are to one another as the squares on their
diameters, afterwards using this proposition to prove that
similar segments are to one another as the squares on their
bases. Euclid of course proves XII. 2 by the method of
exhaustion, the invention of which is attributed to Eudoxus
on the ground of notices in Archimedes. 1 This method
depends on the use of a certain lemma known as the Axiom
of Archimedes, or, alternatively, a lemma similar to it. The
lemma used by Euclid is his proposition X. 1, which is closely
related to Archimedes's lemma in that the latter is practically
used in the proof of it. Unfortunately we have no infor-
mation as to the nature of Hippocrates's proof; if, however,
it amounted to a genuine proof, as Eudemus seems to imply,
it is difficult to see how it could have been effected other-
wise than by some anticipation in essence of the method of
exhaustion.
Theodorus of Cyrene, who is mentioned by Proclus along
with Hippocrates as a celebrated geometer and is claimed by
Iamblichus as a Pythagorean, 2 is only known to us from
Plato's Theaetetus. He is said to have been Plato's teacher
in mathematics, 3 and it is likely enough that Plato, while on
his way to or from Egypt, spent some time with Theodorus at
Cyrene, 4 though, as we gather from the Theaetetus, Theodorus
had also been in Athens in the time of Socrates. We learn
from the same dialogue that he was a pupil of Protagoras, and
was distinguished not only in geometry but in astronomy,
arithmetic, music, and all educational subjects. 5 The one notice
1 Prefaces to On the Sphere and Cylinder, i, and Quadrature of the
Parabola.
2 Iambi. Vit. Pyth. c. 86. 3 Diog. L. ii. 103.
4 Cf. Diog. L. iii. 6.
B Plato, Theaetetus, 161 B, 162 A ; ib. 145 a, c, d.
THEODORUS OF CYRENE 203
whicli wo have of a particular achievement of his suggests thai-
it was he who first carried the theory of irrationals beyond
the first step, namely the discovery by the Pythagoreans
of the irrationality of y/2. According to the Theaetetus, 1
Theodoras
1 was proving 2 to us a certain thing about square roots
(Svvdfieis), I mean (the square roots, i.e. sides) of three square
feet and of five square feet, namely that these roots are not
commensurable in length with the foot-length, and he went on
in this way, taking all the separate cases up to the root of«
1 7 square feet, at which point, for some reason, he stopped '.
That is, he proved the irrationality of \/3, \^5 ... up to
V 17. It does not appear, however, that he had reached any
definition of a surd in general or proved any general proposi-
tion about all surds, for Theaetetus goes on to say :
' The idea occurred to the two of us (Theaetetus and the
younger Socrates), seeing that these square roots appeared
1 Theaetetus, 147 d sq.
2 Ilepl bwdpeiav n rjpiv Qe68a>pos ode eypacpe, ttJs re rptnoBos wept Ka\
7r€VTeTro8os [a7ro0atVcoi/] otl p.r)Kei ov avpperpoi. Tfj 7ro$ialq. Certain writers
(H. Vogt in particular) persist in taking eypa<pe in this sentence to mean
drew or constructed. The idea is that Theodorus's exposition must have
included two things, first the construction of straight lines representing
\/3, \/5 ... (of course by means of the Pythagorean theorem, Eucl. I. 47),
in order to show that these straight lines exist, and secondly the proof
that each of them is incommensurable with 1 ; therefore, it is argued,
eyparfx; must indicate the construction and dnocpaivcop the proof. But in
the first place it is impossible that eypacf)* tl nepi, ' he wrote something
about ' (roots), should mean ' constructed each of the roots '. Moreover, if
dncxfiaivcov is bracketed (as it is by Burnet), the supposed contrast between
eypacpe and cnrofyaivw disappears, and eypacpe must mean 'proved 1 , in
accordance with the natural meaning of eypacpe ti, because there is
nothing else to govern on prjKei, ktA. (' that they are not commensurable
in length . . .'), which phrase is of course a closer description of n. There
are plenty of instances of ypdfaiv in the sense of ' prove '. Aristotle says
(Topics, e. 3, 158 b 29) 'It would appear that in mathematics too some
things are difficult to prove (ov paSuos- ypd(peo-6ai) owing to the want of
a definition, e.g. that a straight line parallel to the side and cutting a plane
figure (parallelogram) divides the straight line (side) and the area simi-
larly '. Cf. Archimedes, On the Sphere and Cylinder, ii, Pref., ' It happens
that most of them are proved (ypd(peo-dai) by means of the theorems . . . ' ;
' Such of the theorems and problems as are proved (ypdfaTai) by means of
these theorems I have proved (or written out, ypu\f/as) and send you
in this book ' ; Quadrature of a Parabola, Pref., ' I have proved (eypa<j>ov)
that every cone is one third of the cylinder with the same base and equal
height by assuming a lemma similar to that aforesaid.'
I do not deny that Theodorus constructed his ' roots ' ; I have no doubt
that he did ; but this is not what eypa<fie tl means.
204 THE ELEMENTS DOWN TO PLATO'S TIME
to be unlimited in multitude, to try to arrive at one collective
term by which we could designate all these roots . . . We
divided number in general into two classes. The number
which can be expressed as equal multiplied by equal (icroy
Io-clkis) we likened to a square in form, and we called it
square and equilateral (la-oirXevpov) . . . The intermediate
number, such as three, five, and any number which cannot
be expressed as equal multiplied by equal, but is either less
times more or more times less, so that it is always contained
by a greater and a less side, we likened to an oblong figure
(TTpofjLrJKei a-yrjfiaTL) and called an oblong number. . . . Such
straight lines then as square the equilateral and plane number
we defined as length (firJKos), and such as square the oblong
(we called) square roots (Svvdjieis) as not being commensurable
with the others in length but only in the plane areas to which
their squares are equal. And there is another distinction of
the same sort with regard to solids/
Plato gives no hint as to how Theodoras proved the proposi-
tions attributed to him, namely that \/3, V5 ... V 17 are
all incommensurable with 1 ; there is therefore a wide field
open for speculation, and several conjectures have been put
forward.
(1) Hultsch, in a paper on Archimedes's approximations to
square roots, suggested that Theodoras took the line of seeking
successive approximations. Just as | , the first approximation
to a/2, was obtained by putting 2 = ■§■§■, Theodoras might
have started from 3 = ff, and found J or \\\ as a first
approximation, and then, seeing that \\\ > v / 3>l|, might
(by successive trials, probably) have found that
111 i _i 1 ^ ./o s il 1 JL_ ^i i_
1 2 8 16 32 64 x v,7 ^ ± 2 8 16 32 128*
But the method of finding closer and closer approximations,
although it might afford a presumption that the true value
cannot be exactly expressed in fractions, would leave Theodoras
as far as ever from proving that a/ 3 is incommensurable.
(2) There is no mention of V2 in our passage, and Theodoras
probably omitted this case because the incommensurability
of \/2 and the traditional method of proving it were already
known. The traditional proof was, as we have seen, a reduetio
ad absurdum showing that, if v/2 is commensurable with 1,
it will follow that the same number is both even and odd,
i.e. both divisible and not divisible by 2. The same method
• THEODORUS OF CYRENE 205
of proof can be adapted to the cases of \^S, </5, &c, if 3, 5 ...
are substituted for 2 in the proof; e.g. we can prove that,
if ^/3 is commensurable with 1, then the same number will
be both divisible and not divisible by 3. One suggestion,
therefore, is that Theodorus may have applied this method
to all the cases from a/3 to VI 7. We can put the proof
quite generally thus. Suppose that JV is a non-square number
such as 3, 5 ..., and, if possible, let VN = m/n, where m, n
are integers prime to one another.
Therefore m 2 = N .n 2 ;
therefore m 2 is divisible by N, so that m also is a multiple
of N.
Let im = ijl.N, (1)
and consequently n 2 = iV./r*.
Then in the same way we can prove that n is a multiple
of JV.
Let n = v.N (2)
It follows from (1) and (2) that m/n = p/v, where /jl < m
and v < n ; therefore m/n is not in its lowest terms, which
is contrary to the hypothesis.
The objection to this conjecture as to the nature of
Theodorus's proof is that it is so easy an adaptation of the
traditional proof regarding V2 that it would hardly be
important enough to mention as a new discovery. Also it
would be quite unnecessary to repeat the proof for every
case up to V 1 7 ; for it would be clear, long before V 1 7 was
reached, that it is generally applicable. The latter objection
seems to me to have force. The former objection may or may
not ; for I do not feel sure that Plato is necessarily attributing
any important new discovery to Theodorus. The object of
the whole context is to show that a definition by mere
enumeration is no definition; e.g. it is no definition of kiri-
(TTrjfjLT] to enumerate particular kwLo-Trjuai (as shoemaking,
carpentering, and the like) ; this is to put the cart before the
horse, the general definition of e7ncrTrjfjLrj being logically prior.
Hence it was probably Theaetetus's generalization of the
procedure of Theodorus which impressed Plato as being
original and important rather than Theodorus's proofs them-
selves.
206 THE ELEMENTS DOWN TO PLATO'S TIME
(3) The third hypothesis is that of Zeuthen. 1 He starts
with the assumptions (a) that the method of proof used by
Theodoras must have been original enough to call for special
notice from Plato, and (b) that it must have been of such
a kind that the application of it to each surd required to be
set out separately in consequence of the variations in the
numbers entering into the proofs. Neither of these con-
ditions is satisfied by the hypothesis of a mere adaptation to
\/3, \/5 ... of the traditional proof with regard to V2.
Zeuthen therefore suggests another hypothesis as satisfying
both conditions, namely that Theodoras used the criterion
furnished by the process of finding the greatest common
measure as stated in the theorem of Eucl. X. 2. 'If, when
the lesser of two unequal magnitudes is continually subtracted
in turn from the greater [this includes the subtraction
from any term of the highest multiple of another that it
contains], that which is left never measures the one before
it, the magnitudes will be incommensurable ' ; that is, if two
magnitudes are such that the process of finding their G. C. M.
never comes to an end, the two magnitudes are incommensur-
able. True, the proposition Eucl. X. 2 depends on the famous
X. 1 (Given two unequal magnitudes, if from the greater
there be subtracted more than the half (or the half), from the
remainder more than the half (or the half), and so on, there
will be left, ultimately, some magnitude less than the lesser
of the original magnitudes), which is based on the famous
postulate of Eudoxus (= Eucl. V, Def. 4), and therefore belongs
to a later date. Zeuthen gets over this objection by pointing
out that the necessity of X. 1 for a rigorous demonstration
of X. 2 may not have been noticed at the time; Theodoras
may have proceeded by intuition, or he may even have
postulated the truth proved in X. 1.
The most obvious case in which incommensurability can be
proved by using the process of finding the greatest common
measure is that of the two segments of a straight line divided
in extreme and mean ratio. For, if A B is divided in this way
at C, we have only to mark off along CA (the greater segment)
1 Zeuthen, ' Sur la constitution des livres arithmetiques des Elements
d'Euclide et leur rapport a la question de rirrationalite ' in Oversigt over
det kgl. Danske videnskabemes Selskabs Forhandlinger, 1915, pp. 422 sq.
THEODORUS OF GYRENE 207
a length CD equal to CB (the lesser segment), and C A is then
divided at D in extreme and mean ratio, CD being the
preater segment. (Eucl. XIII. 5 is the equivalent of this
A* D E C B
I 1 1 1 1
proposition.) Similarly, DC is so divided if we set oft* DE
along it equal to DA ; and so on. This is precisely the
process of finding the greatest common measure of AC, CB,
the quotient being always unity ; and the process never comes
to an end. Therefore AC, CB are incommensurable. What
is proved in this case is the irrationality of J(\/5 — 1). This
of course shows incidentally that \/5 is incommensurable
with 1. It has been suggested, in view of the easiness of the
above proof, that the irrational may first have been discovered
with reference to the segments of a straight line cut in extreme
and mean ratio, rather than with reference to the diagonal
of a square in relation to its side. But this seems, on the
whole, improbable.
Theodorus would, of course, give a geometrical form to the
process of finding the G. C. M., after he had represented in
a figure the particular surd which he was investigating.
Zeuthen illustrates by two cases, v 7 5 and V 3.
We will take the former, which is the easier. The process
of finding the G. C. M. (if any) of V 5 and 1 is as follows :
1) V5(2
2
^5-2)1 (4
4(\/5-2)
(n/5-2) 2
[The explanation of the second division is this :
1 = ( v / 5-2)(v / 5 + 2) = 4 (a/5 -2) + (</5-2) 2 .]
Since, then, the ratio of the last term ( \f 5 — 2) 2 to the pre-
ceding one, >/5 — 2, is the same as the ratio of \/5— 2 to I,
the process will never end.
Zeuthen has a geometrical proof which is not difficult ; but
I think the following proof is neater and easier.
Let A43C be a triangle right-angled at B, suc*h that AB = 1,
BC = 2, and therefore AC = Vs.
208 THE ELEMENTS DOWN TO PLATO'S TIME
Cut off CD from G A equal to GB, and draw DE at right
angles to GA. Then DE = EB.
Now AD — V5 — 2, and by similar triangles
DE= 2AD= 2(a/5-2).
Cut off from EA the portion EF equal to
ED, and draw FG at right angles to A E.
Then AF= AB-BF= AB-2DE
= 1-4(^5-2)
= (v/5-2) 2 .
Therefore AJ9G f , ADE, AFG are diminishing
similar triangles such that
AB:AD:AF=l:(y5-2):(V5-2) 2 ,
and so on.
Also AB > FB, i.e. 2DE or 4 AD.
Therefore the side of each triangle in the series is less than
•J of the corresponding side of the preceding triangle.
In the case of \/3 the process of finding the G. C. M. of
V 3 and 1 gives
\ f
1 ) v/3 ( 1
1
</3-l) 1
a/3
(1
J(V8-1)«)-/S-1 (2
(y/3-1) 2
i(VB-l)»
the ratio of i(\/3 — l) 2 to |(\/3 — l) 3 being the same as that
of 1 to (\/3-l).
This case is more difficult to show in geometrical form
because we have to make one more
division before recurrence takes place.
The cases V 1 and V 17 are exactly
similar to that of -/5.
The irrationality of V2 can, of course,
be proved by the same method. If ABGD
is a square, we mark off along the diagonal
AG a, length AE equal to A B and draw
EF at right angles to AG. The same
thing is then done with the triangle GEF
THEODORUS OF CYRENE 209
as with the triangle ABC, and so on. This could not have
escaped Theodoras if his proof in the cases of \/3, V5 ...
took the form suggested by Zeuthen ; but he was presumably
content to accept the traditional proof with regard to V 2.
The conjecture of Zeuthen is very ingenious, but, as he
admits, it necessarily remains a hypothesis.
Theaetetus 1 (about 415-369 B.C.) made important contribu-
tions to the body of the Elements. These related to two
subjects in particular, (a) the theory of irrationals, and (b) the
five regular solids.
That Theaetetus actually succeeded in generalizing the
theory of irrationals on the lines indicated in the second part
of the passage from Plato's dialogue is confirmed by other
evidence. The commentary on Eucl. X, which has survived
in Arabic and is attributed to Pappus, says (in the passage
partly quoted above, p. 155) that the theory of irrationals
'had its origin in the school of Pythagoras. It was con-
siderably developed by Theaetetus the Athenian, who gave
proof in this part of mathematics, as in others, of ability
which has been justly admired. ... As for the exact dis-
tinctions of the above-named magnitudes and the rigorous
demonstrations of the propositions to which this theory gives
rise, I believe that they were chiefly established by this
mathematician. For Theaetetus had distinguished square
roots 2 commensurable in length from those which are incom-
mensurable, and had divided the well-known species of
irrational lines after the different means, assigning the medial
to geometry, the binomial to arithmetic, and the apotome to
harmony, as is stated by Eudemus the Peripatetic.' 3
1 On Theaetetus the reader may consult a recent dissertation, De Theae-
teto Atheniensi mathematico, by Eva Sachs (Berlin, 1914).
2 ' Square roots '. The word in Woepcke's translation is ' puissances ',
which indicates that the original word was Swa^tis. This word is always
ambiguous ; it might mean ' squares ', but I have translated it ' square
roots ' because the 8vvafxis of Theaetetus's definition is undoubtedly the
square root of a non-square number, a surd. The distinction in that case
would appear to be between ' square roots ' commensurable in length and
square roots commensurable in square only ; thus ^3 and a/12 are
commensurable in length, while a/3 and a/7 are commensurable in
square only. I do not see how dwdpeis could here mean squares ; for
' squares commensurable in length ' is not an intelligible phrase, and it
does not seem legitimate to expand it into ' squares (on straight lines)
commensurable in length ',
s For an explanation of this see The Thirteen Boohs of Euclid's Elements
vol. iii, p. 4.
1523 . P
210 THE ELEMENTS DOWN TO PLATO'S TIME
The irrationals called by the names here italicized are
described in Eucl. X. 21, 36 and 73 respectively.
Again, a scholiast 1 on Eucl. X. 9 (containing the general
theorem that squares which have not to one another the ratio
of a square number to a square number have their sides
incommensurable in length) definitely attributes the discovery
of this theorem to Theaetetus. But, in accordance with the
traditional practice in Greek geometry, it was necessary to
prove the existence of such incommensurable ratios, and this
is done in the porism to Eucl. X. 6 by a geometrical con-
struction ; the porism first states that, given a straight line a
and any two numbers m, n, we can find a straight line x such
that a : x = m:n; next it is shown that, if y be taken a mean
proportional between a and x, then
a 2 : y 2 = a : x — m : n ;
if, therefore, the ratio m : n is not a ratio of a square to
a square, we have constructed an irrational straight line
a V(n/m) and therefore shown that such a straight line
exists.
The proof of Eucl. X. 9 formally depends on VIII. 1 1 alone
(to the effect that between two square numbers there is one
mean proportional number, and the square has to the square
the duplicate ratio of that which the side has to the side) ;
and VIII. 11 again depends on VII. 17 and 18 (to the effect
that ab : ac = b : c, and a:b = ac.bc, propositions which are
not identical). But Zeuthen points out that these propositions
are an inseparable part of a whole theory established in
Book VII and the early part of Book VIII, and that the
real demonstration of X. 9 is rather contained in propositions
of these Books which give a rigorous proof of the necessary
and sufficient conditions for the rationality of the square
roots of numerical fractions and integral numbers, notably
VII. 27 and the propositions leading up to it, as well as
VIII. 2. He therefore suggests that the theory established
in the early part of Book VII was not due to the Pytha-
goreans, but was an innovation made by Theaetetus with the
direct object of laying down a scientific basis for his theory
of irrationals, and that this, rather than the mere formulation
1 X, No. 62 (Heiberg's Euclid, vol. v, p. 450).
THEAETETUS 211
of the theorem of Eucl. X. 9, was the achievement which Plato
intended to hold up to admiration.
This conjecture is of great interest, but it is, so far as
I know, without any positive confirmation. On the other
hand, there are circumstances which suggest doubts. For
example, Zeuthen himself admits that Hippocrates, who re-
duced the duplication of the cube to the rinding of two mean
proportionals, must have had a proposition corresponding to
the very proposition VIII. 11 on which X. 9 formally depends.
Secondly, in the extract from Simplicius about the squaring
of lunes by Hippocrates, we have seen that the proportionality
of similar segments of circles to the circles of which they form
part is explained by the statement that ' similar segments are
those which are the same part of the circles ' ; and if we may
take this to be a quotation by Eudemus from Hippocrates's
own argument, the inference is that Hippocrates had a defini-
tion of numerical proportion which was at all events near
to that of Eucl. VII, Def. 20. Thirdly, there is the proof
(presently to be given) by Archytas of the proposition that
there can be no number which is a (geometric) mean between
two consecutive integral numbers, in which proof it will
be seen that several propositions of Eucl., Book VII, are
pre-supposed ; but Archytas lived (say) 430-S65 B.C., and
Theaetetus was some years younger. I am not, therefore,
prepared to give up the view, whicli has hitherto found
general acceptance, that the Pythagoreans already had a
theory of proportion of a numerical kind on the lines, though
not necessarily or even probably with anything like the
fullness and elaboration, of Eucl., Book VII.
While Pappus, in the commentary quoted, says that Theae-
tetus distinguished the well-known species of irrationals, and
in particular the medial, the binomial, and the apotome, he
proceeds thus :
' As for Euclid, he set himself to give rigorous rules, which
he established, relative to commensurability and incommen-
surability in general ; he made precise the definitions and
distinctions between rational and irrational magnitudes, he
set out a great number of orders of irrational magnitudes,
and finally he made clear their whole extent.'
As Euclid proves that there are thirteen irrational straight
p2
212 THE ELEMENTS DOWN TO PLATO'S TIME
lines in all, we may perhaps assume that the subdivision of
the three species of irrationals distinguished by Theaetetus
into thirteen was due to Euclid himself, while the last words
of the quotation seem to refer to Eucl. X. 115, where it is
proved that from the medial straight line an unlimited number
of other irrationals can be derived which are all different from
it and from one another.
It will be remembered that, at the end of the passage of the
Theaetetus containing the definition of ' square roots ' or surds,
Theaetetus says that ' there is a similar distinction in the case
of solids '. We know nothing of any further development
of a theory of irrationals arising from solids ; but Theaetetus
doubtless had in mind a distinction related to VIII. 12 (the
theorem that between two cube numbers there are two mean
proportional numbers) in the same way as the definition of
a ' square root ' or surd is related to VIII. 1 1 ; that is to say,
he referred to the incommensurable cube root of a non-cube
number which is the product of three factors.
Besides laying the foundation of the theory of irrationals
as we find it in Eucl., Book X, Theaetetus contributed no less
substantially to another portion of the Elements, namely
Book XIII, which is devoted (after twelve introductory
propositions) to constructing the five regular solids, circum-
scribing spheres about them, and finding the relation between
the dimensions of the respective solids and the circumscribing
spheres. We have already mentioned (pp. 159, 162) the tradi-
tions that Theaetetus was the first to ' construct' or 'write upon'
the five regular solids, 1 and that his name was specially
associated with the octahedron and the icosahedron. 2 There
can be little doubt that Theaetetus's ' construction ' of, or
treatise upon, the regular solids gave the theoretical con-
structions much as we find them in Euclid.
Of the mathematicians of Plato's time, two others are
mentioned with Theaetetus as having increased the number
of theorems in geometry and made a further advance towards
a scientific grouping of them, Leodamas of Thasos and
Arch yt as of Tar as. With regard to the former we are
1 Sllidas, S.V. 0ennv?TO9.
2 Schol. 1 to Eucl. XIII (Euclid, ed. Heiberg, vol. v, p. 654).
ARCHYTAS 213
told that Plato ' explained (elo-rjyrjo-aTo) to Leodamas of Thasos
the method of inquiry by analysis ' 1 ; Proclus's account is
fuller, stating that the finest method for discovering lemmas
in geometry is that ' which by means of analysis carries the
thing sought up to an acknowledged principle, a method
which Plato, as they say, communicated to Leodamas, and
by which the latter too is said to have discovered many
things in geometry \ 2 Nothing more than this is known of
Leodamas, but the passages are noteworthy as having given
rise to the idea that Plato invented the method of mathe-
matical analysis, an idea which, as we shall see later on, seems
nevertheless to be based on a misapprehension.
Archytas of Taras, a Pythagorean, the friend of Plato,
nourished in the first half of the fourth century, say 400 to
365 B.C. Plato made his acquaintance when staying in Magna
Graecia, and he is said, by means of a letter, to have saved
Plato from death at the hands of Dionysius. Statesman and
philosopher, he was famed for every sort of accomplishment.
He was general of the forces of his city-state for seven years,
though ordinarily the law forbade any one to hold the post
for more than a year; and he was never beaten. He is
said to have been the first to write a systematic treatise on
mechanics based on mathematical principles/ 5 Vitruvius men-
tions that, like Archimedes, Ctesibius, Nymphodorus, and
Philo of Byzantium, Archytas wrote on machines 4 ; two
mechanical devices in particular are attributed to him, one
a mechanical dove made of wood which would fly, 5 the
other a rattle which, according to Aristotle, was found useful
to ' give to children to occupy them, and so prevent them
from breaking things about the house (for the young are
incapable of keeping still) '. 6
We have already seen Archytas distinguishing the four
mathematical sciences, geometry, arithmetic, sphaeric (or
astronomy), and music, comparing the art of calculation with
geometry in respect of its relative efficiency and conclusive-'
ness, and defining the three means in music, the arithmetic,
1 Diog. L. iii. 24. 2 Proclus on Eucl. I, p. 211. 19-23.
3 Diog. L. viii. 79-83.
4 Vitruvius, Be architectural Praef. vii. 14.
5 Gellius, x. 12. 8, after Favorinus {Vors. i 3 , p. 325. 21-9).
6 Aristotle, Politics, E (0). 6, 1340 b 26.
214 THE ELEMENTS DOWN TO PLATO'S TIME
the geometric, and the harmonic (a name substituted by
Archytas and Hippasus tor the older name 'sub-contrary').
From his mention of sphaeric in connexion with his state-
ment that ' the mathematicians have given us clear knowledge
about the speed of the heavenly bodies and their risings and
settings ' we gather that in Archytas's time astronomy was
already treated mathematically, the properties of the sphere
being studied so far as necessary to explain the movements
in the celestial sphere. He discussed too the question whether
the universe is unlimited in extent, using the following
argument.
' If I were at the outside, say at the heaven of the fixed
stars, could I stretch my hand or my stick outwards or not ?
To suppose that I could not is absurd ; and if I can stretch
it out, that which is outside must be either body or space (it
makes no difference which it is, as we shall see). We may
then in the same way get to the outside of that again, and
so on, asking on arrival at each new limit the same question ;
and if there is always a new place to which the stick may be
held out, this clearly involves extension without limit. If
now what so extends is body, the proposition is proved ; but
even if it is space, then, since space is that in which body
is or can be, and in the case of eternal things we must treat
that which potentially is as being, it follows equally that there
must be body and space (extending) without limit.' l
In geometry, while Archytas doubtless increased the number
of theorems (as Proclus says), only one fragment of his has
survived, namely the solution of the problem of finding two
mean proportionals (equivalent to the duplication of the cube)
by a remarkable theoretical construction in three dimensions.
As this, however, belongs to higher geometry and not to the
Elements, the description of it will come more appropriately
in another place (pp. 246-9).
In music he gave the numerical ratios representing the
intervals of the tetrachord on three scales, the anharmonic,
the chromatic, and the diatonic. 2 He held that sound is due
to impact, and that higher tones correspond to quicker motion
communicated to the air, and lower tones to slower motion. 3
1 Simplicius in Phys., p. 467. 26. 2 Ptol. harm. i. 13, p. 31 Wall.
3 Porph. in Ptol. harm., p. 236 (Vors. i 3 , p. 232-3); Theon of Smyrna,
p. 61. 11-17.
ARCHYTAS 215
Of the fragments of Archytas handed down to us the most
interesting from the point of view of this chapter is a proof
of the proposition that there can be no number which is
a (geometric) mean between two numbers in the ratio known
as eTTi/jiopios or supevparticularis, that is, (n+l):n. This
proof is preserved by Boetius l , and the noteworthy fact about
it is that it is substantially identical with the proof of the
same theorem in Prop. 3 of Euclid's tract on the Sectio
canonist I will quote Archytas's proof in full, in order to
show the slight differences from the Euclidean form and
notation.
Let A, B be the given ' superparticularis proportio ' (kiri-
IxopLov Stdo-rrjfia in Euclid). [Archytas writes the smaller
number first (instead of second, as Euclid does) ; we are then
to suppose that A, B are integral numbers in the ratio of
n to (n+ 1). |
Take C, BE the smallest numbers which are in the ratio
of A to B. [Here BE means B + E] in this respect the
notation differs from that of Euclid, who, as usual, takes
a straight line BF divided into two parts at G, the parts
BG, GF corresponding to the B and E respectively in
Archytas's proof. The step of finding C, BE the smallest
numbers in the same ratio as that of A to B presupposes
Eucl. VII. 33 applied to two numbers.]
Then BE exceeds G by an aliquot part of itself and of C
[cf. the definition of kirifiopLos dpcOfios in Nicomachus, i. 19. l].
Let B be the excess [i.e. we suppose E equal to 0].
I say that B is not a number, but a unit.
For, if B is a number and an aliquot part of BE, it measures
BE; therefore it measures E, that is, G.
Thus B measures both G and BE: which is impossible,
since the smallest numbers which are in the same ratio as
any numbers are prime to one another. [This presupposes
Eucl. VII. 22.]
Therefore B is a unit ; that is, BE exceeds C by a unit.
Hence no number can be found which is a mean between
the two numbers C, BE [for there is no integer intervening].
1 Boetius, De inst. mus. iii. 11, pp. 285-6 Friedlein.
2 Musici scriptores Graeci, ed. Jan, p. 14 ; Heiberg and Menge's Euclid,
vol. viii, p. 162.
216 THE ELEMENTS DOWN TO PLATO'S TIME
Therefore neither can any number be a mean between the
original numbers A, B, which are in the same ratio as C, BE
[cf. the more general proposition, Eucl. VIII. 8 ; the particular
inference is a consequence of Eucl. VII. 20, to the effect that
the least numbers of those which have the same ratio with
them measure the latter the same number of times, the greater
the greater and the less the less].
Since this proof cites as known several propositions corre-
sponding to propositions in Euclid, Book VII, it affords a strong
presumption that there already existed, at least as early as
the time of Arcbytas, a treatise of some sort on the Elements
of Arithmetic in a form similar to the Euclidean, and con-
taining many of the propositions afterwards embodied by
Euclid in his arithmetical books.
Summary.
We are now in a position to form an idea of the scope of
the Elements at the stage which they had reached in Plato's
time. The substance of Eucl. I-IV was practically complete.
Book V was of course missing, because the theory of proportion
elaborated in that book was the creation of Eudoxus. The
Pythagoreans had a theory of proportion applicable to com-
mensurable magnitudes only ; this was probably a numerical
theory on lines similar to those of Eucl., Book VII. But the
theorems of Eucl., Book VI, in general, albeit insufficiently
established in so far as they depended on the numerical theory
of proportion, were known and used by the Pythagoreans.
We have seen reason to suppose that there existed Elements
of Arithmetic partly (at all events) on the lines of Eucl.,
Book VII, while some propositions of Book VIII (e.g. Props.
11 and 12) were also common property. The Pythagoreans,
too, conceived the idea of perfect numbers (numbers equal to
the sum of all their divisors) if they had not actually shown
(as Euclid does in IX. 36) how they are evolved. There can
also be little doubt that many of the properties of plane and
solid numbers and of similar numbers of both classes proved in
Euclid, Books VIII and IX, were known before Plato's time.
We come next to Book X, and it is plain that the foundation
of the whole had been well and truly laid by Theaetetus, and
SUMMARY 217
the main varieties of irrationals distinguished, though their
classification was not carried so far as in Euclid.
The substance of Book XI. 1—19 must already have been in-
cluded in the Elements (e.g. Eucl. XI. 19 is assumed in Archytas's
construction for the two mean proportionals), and the whole
theory of the section of Book XI in question would be required
for Theaetetus's work on the five regular solids: XI. 21 must
have been known to the Pythagoreans : while there is nothing
in the latter portion of the book about parallelepipedal solids
which (subject to the want of a rigorous theory of proportion)
was not within the powers of those who were familiar with
the theory of plane and solid numbers.
Book XII employs throughout the method of exhaustion,
the orthodox form of which is attributed to Eudoxus, who
grounded it upon a lemma known as Archimedes's Axiom or
its equivalent (Eucl. X. 1). Yet even XII. 2, to the effect that
circles are to one another as the square of their diameters, had
already been anticipated by Hippocrates of Chios, while
Democritus had discovered the truth of the theorems of
XII. 7, Por., about the volume of a pyramid, and XII. 10,
about the volume of a cone.
As in the case of Book X, it would appear that Euclid was
indebted to Theaetetus for much of the substance of Book XIII,
the latter part of which (Props. 12-18) is devoted to the
construction of the five regular solids, and the inscribing of
them in spheres.
There is therefore probably little in the whole compass of
the Elements of Euclid, except the new theory of proportion due
to Eudoxus and its consequences, which was not in substance
included in the recognized content of geometry and arithmetic
by Plato's time, although the form and arrangement of the
subject-matter and the methods employed in particular cases
were different from what we find in Euclid.
VII
SPECIAL PROBLEMS
Simultaneously with the gradual evolution of the Elements,
the Greeks were occupying themselves with problems in
higher geometry; three problems in particular, the squaring
of the circle, the doubling of the cube, and the trisection of
any given angle, were rallying-points for mathematicians
during three centuries at least, and the whole course of Greek
geometry was profoundly influenced by the character of the
specialized investigations which had their origin in the attempts
to solve these problems. In illustration we need only refer
to the subject of conic sections which began with the use
made of two of the curves for the finding of two mean pro-
portionals.
The Greeks classified problems according to the means by
which they were solved. The ancients, says Pappus, divided
them into three classes, which they called plane, solid, and
linear respectively. Problems were plane if they could be
solved by means of the straight line and circle only, solid
if they could be solved by means of one or more conic sections,
and linear if their solution required the use of other curves
still more complicated and difficult to construct, such as spirals,
quadratrices, cochloids (conchoids) and cissoids, or again the
various curves included in the class of ' loci on surfaces ' (tottol
7T/00? e7TL(f)av€LaL9), as they were called. 1 There was a corre-
sponding distinction between loci : plane loci are straight
lines or circles; solid loci are, according to the most strict
classification, conies only, which arise from the sections of
certain solids, namely cones ; while linear loci include all
1 Pappus, iii, pp. 54-6, iv, pp. 270-2.
CLASSIFICATION OF PROBLEMS 219
higher curves. 1 Another classification of loci divides them
into loci on lines (tottol rrpbs ypauuais) and loci on surfaces
(tottol rrpbs kTricfxiveLais;)! 1 The former term is found in
Proclus, and seems to be used in the sense both of loci which
are lines (including of course curves) and of loci which are
spaces bounded by lines; e.g. Proclus speaks of 'the whole
space between the parallels' in Eucl. I. 35 as being the locus
of the (equal) parallelograms 'on the same base and in the
same parallels'. 3 Similarly loci on surfaces in Proclus may
be loci which are surfaces; but Pappus, who gives lemmas
to the two books of Euclid under that title, seems to imply
that they were curves drawn on surfaces, e.g. the cylindrical
helix. 4
It is evident that the Greek geometers came very early
to the conclusion that the three problems in question were not
'plane, but required for their solution either higher curves
than circles or constructions more mechanical in character
than the mere use of the ruler arid compasses in the sense of
Euclid's Postulates 1-3. It was probably about 420 B.C. that
Hippias of Elis invented the curve known as the quadratrix
for the purpose of trisecting any angle, and it was in the first
half of the fourth century that Archytas used for the dupli-
cation of the cube a solid construction involving the revolution
of plane figures in space, one of which made a tore or anchor-
ring with internal diameter nil. There are very few records
of illusory attempts to do the impossible in these cases. It is
practically only in the case of the squaring of the circle that
we read of abortive efforts made by ' plane ' methods, and none
of these (with the possible exception of Bry son's, if the
accounts of his argument are correct) involved any real
fallacy. On the other hand, the bold pronouncement of
Antiphon the Sophist that by inscribing in a circle a series
of regular polygons each of which has twice as many sides
as the preceding one, we shall use up or exhaust the area of
the circle, though it was in advance of his time and was
condemned as a fallacy on the technical ground that a straight
line cannot coincide with an arc of a circle however short
its length, contained an idea destined to be fruitful in the
1 Cf. Pappus, vii, p. 662, 10-15. 2 Proclus on Eucl. I, p. 394. 19.
3 lb., p. 395. 5. " * Pappus, iv, p. 258 sq.
220 SPECIAL PROBLEMS
hands of later and abler geometers, since it gives a method
of approximating, with any desired degree of accuracy, to the
area of a circle, and lies at the root of the method of exhaustion
as established by Eudoxus. As regards Hippocrates's quadra-
ture of lunes, we must, notwithstanding the criticism of
Aristotle charging him with a paralogism, decline to believe
that he was under any illusion as to the limits of what his
method could accomplish, or thought that he had actually
squared the circle.
The squaring of the circle.
There is presumably no problem which has exercised such
a fascination throughout the ages as that of rectifying or
squaring the circle ; and it is a curious fact that its attraction
has been no less (perhaps even greater) for the non-mathe-
matician than for the mathematician. It was naturally the
kind of problem which the Greeks, of all people, would take
up with zest the moment that its difficulty was realized. The
first name connected with the problem is Anaxagoras, who
is said to have occupied himself with it when in prison. 1
The Pythagoreans claimed that it was solved in their school,
' as is clear from the demonstrations of Sextus the Pythagorean,
who got his method of demonstration from early tradition ' 2 ;
but Sextus, or rather Sextius, lived in the reign of Augustus
or Tiberius, and, for the usual reasons, no value can be
attached to the statement.
The first serious attempts to solve the problem belong to
the second half of the fifth century B.C. A passage of
Aristophanes's Birds is quoted as evidence of the popularity
of the problem at the time (414 B.C.) of its first representation.
Aristophanes introduces Meton, the astronomer and discoverer
of the Metonic cycle of 19 years, who brings with him a ruler
and compasses, and makes a certain construction ' in order that
your circle may become square \ 3 This is a play upon words,
because what Meton really does is to divide a circle into four
quadrants by two diameters at right angles to one another ;
the idea is of streets radiating from the agora in the centre
1 Plutarch, De exil. 17, p. 607 F.
2 Iambi, aj). Simpl. in Categ., p. 192, 16-19 K., 64 b 11 Brandis.
3 Aristophanes, Birds 1005.
THE SQUARING OF THE CIRCLE
221
of a town ; the word Terpdycovos then really means ' with four
(right) angles ' (at the centre), and not ' square ', but the word
conveys a laughing allusion to the problem of squaring all
the same.
We have already given an account of Hippocrates's quadra-
tures of lunes. These formed a sort of "prolvbsio, and clearly
did not purport to be a solution of the problem ; Hippocrates
was aware that ' plane ' methods would not solve it, but, as
a matter of interest, he wished to show that, if circles could
not be squared by these methods, they could be employed
to find the area of some figures bounded by arcs of circles,
namely certain lunes, and even of the sum of a certain circle
and a certain lune.
Antiphon of Athens, the Sophist and a contemporary of
Socrates, is the next person to claim attention. We owe
to Aristotle and his commentators our knowledge of Anti-
phon's method. Aristotle observes that a geometer is only
concerned to refute any fallacious arguments that may be
propounded in his subject if they are based upon the admitted
principles of geometry ; if they are not so based, he is not
concerned to refute them :
' thus it is the geometer's business to refute the quadrature by
means of segments, but it is not his business to refute that
of Antiphon \ 1
As we have seen, the quadrature ' by means of segments ' is
probably Hippocrates's quad-
rature of lunes. Antiphon' s
method is indicated by Themis-
tius 2 and Simplicius. 3 Suppose
there is any regular polygon
inscribed in a circle, e.g. a square
or an equilateral triangle. (Ac-
cording to Themistius, Antiphon
began with an equilateral triangle,
and this seems to be the authentic
version ; Simplicius says he in-
scribed some one of the regular polygons which can be inscribed
1 Arist. Pht/s. i. 2, 185 a 14-17.
2 Them, in Phys., p. 4. 2 sq.. Schenkl.
3 Simp], in Phys., p. 54. 20-55. 24, Diels.
222 THE SQUARING OF THE CIRCLE
in a circle, ' suppose, if it so happen, that the inscribed polygon
is a square '.) On each side of the inscribed triangle or square
as base describe an isosceles triangle with its vertex on the
arc of the smaller segment of the circle subtended by the side.
This gives a regular inscribed polygon with double the number
of sides. Repeat the construction with the new polygon, and
we have an inscribed polygon with four times as many sides as
the original polygon had. Continuing the process,
' Antiphon thought that in this way the area (of the circle)
would be used up, and we should some time have a polygon
inscribed in the circle the sides of which would, owing to their
smallness, coincide with the circumference of the circle. And,
as we can make a square equal to any polygon . . . we shall
be in a position to make a square equal to a circle.'
Simplicius tells us that, while according to Alexander the
geometrical principle hereby infringed is the truth that a circle
touches a straight line in one point (only), Eudemus more
correctly said it was the principle that magnitudes are divisible
without limit ; for, if the area of the circle is divisible without
limit, the process described by Antiphon will never result in
using up the whole area, or in making the sides of the polygon
take the position of the actual circumference of the circle.
But the objection to Antiphon's statement is really no more than
verbal ; Euclid uses exactly the same construction in XII. 2,
only he expresses the conclusion in a different way, saying
that, if the process be continued far enough, the small seg-
ments left over will be together less than any assigned area.
Antiphon in effect said the same thing, which again we express
by saying that the circle is the limit of such an inscribed
polygon when the number of its sides is indefinitely increased.
Antiphon therefore deserves an honourable place in the history
of geometry as having originated the idea of exhausting an
area by means of inscribed regular polygons with an ever
increasing number of sides, an idea upon which, as we said,
Eudoxus founded his epoch-making method of exhaustion.
The practical value of Antiphon's construction is illustrated
by Archimedes's treatise on the Measurement of a Circle,
where, by constructing inscribed and circumscribed regular
polygons with 96 sides, Archimedes proves that 3^ > n > 3^£,
the lower limit, ir > 3^f , being obtained by calculating the
ANTIPHON AND BRYSON 223
perimeter of the inscribed polygon of 96 sides, which is
constructed in Antiphon's manner from an inscribed equilateral
triangle. The same construction starting from a square was
likewise the basis of Vieta's expression for 2/77, namely
2
IT
IT
TT
—
—
COS
COS
—
COS
IT
4
8
16
= Vi.Vi(l+ •§) . s/|(l + ^4(1 + Vi)) ... (ad inf.)
Bryson, who came a generation later than Antiphon, being
a pupil of Socrates or of Euclid of Megara, was the author
of another attempted quadrature which is criticized by
Aristotle as ' sophistic ' and ' eristic ' on the ground that it
was based on principles not special to geometry but applicable
equally to other subjects. 1 The commentators give accounts
of Bryson' s argument which are substantially the same, except
that Alexander speaks of squares inscribed and circumscribed
to a circle 2 , while Themistius and Philoponus speak of any
polygons. 3 According to Alexander, Bryson inscribed a square
in a circle and circumscribed another about it, while he also
took a square intermediate between them (Alexander does not
say how constructed) ; then he argued that, as the intermediate
square is less than the outer and greater than the inner, while
the circle is also less than the outer square and greater than
the inner, and as things which are greater and less than the
same things respectively are equal, it follows that the circle is
equal to the intermediate square : upon which Alexander
remarks that not only is the thing assumed applicable to
other things besides geometrical magnitudes, e.g. to numbers,
times, depths of colour, degrees of heat or cold, &c., but it
is also false because (for instance) 8 and 9 are both less than
10 and greater than 7 and yet they are not equal. As regards
the intermediate square (or polygon), some have assumed that
it was the arithmetic mean between the inscribed and circum-
scribed figures, and others that it was the geometric mean.
Both assumptions seem to be due to misunderstanding 4 ; for
1 Arist. An. Post. i. 9, 75 b 40.
2 Alexander on Soph. EL, p. 90. 10-21, Wallies, 306 b 24 sq., Brandis.
3 Them, on An. Post., p. 19. 11-20, Wallies, 211 b 19, Brandis; Philop.
on An. Post., p. 111. 20-114. 17 W., 211 b 30, Brandis.
4 Psellus (11th cent, a.d.) says, 'there are different opinions as to the
224 THE SQUARING OF THE CIRCLE
the ancient commentators do not attribute to Bryson any such
statement, and indeed, to judge by their discussions of different
interpretations, it would seem that tradition was by no means
clear as to what Bryson actually did say. But it seems
important to note that Themistius states (1) that Bryson
declared the circle to be greater than all inscribed, and less
than all circumscribed, polygons, while he also says (2) that
the assumed axiom is true, though not peculiar to geometry.
This suggests a possible explanation of what otherwise seems
to be an absurd argument. Bryson may have multiplied the
number of the sides of both the inscribed and circumscribed
regular polygons as Antiphon did with inscribed polygons;
he may then have argued that, if we continue this process
long enough, we shall have an inscribed and a circumscribed
polygon differing so little in area that, if we can describe
a polygon intermediate between them in area, the circle, which
is also intermediate in area between the inscribed and circum-
scribed polygons, must be equal to the intermediate polygon. 1
If this is the right explanation, Bryson's name by no means
deserves to be banished from histories of Greek mathematics ;
on the contrary, in so far as he suggested the necessity of
considering circumscribed as well as inscribed polygons, he
went a step further than Antiphon; and the importance of
the idea is attested by the fact that, in the regular method
of exhaustion as practised by Archimedes, use is made of both
inscribed and circumscribed figures, and this compression, as it
were, of a circumscribed and an inscribed figure into one so
that they ultimately coincide with one another, and with the
proper method of finding the area of a circle, but that which has found
the most favour is to take the geometric mean between the inscribed and
circumscribed squares'. I am not aware that he quotes Bryson as the
authority for this method, and it gives the inaccurate value n = >y/8 or
2-8284272 Isaac Argyrus (14th cent.) adds to his account of Bryson
the following sentence : ' For the circumscribed square seems to exceed
the circle by the same amount as the inscribed square is exceeded by the
circle.'
1 It is true that, according to Philoponus, Proclus had before him an
explanation of this kind, but rejected it on the ground that it would
mean that the circle must actually be the intermediate polygon and not
only be equal to it, in which case Bryson's contention would be tanta-
mount to Antiphon 's, whereas according to Aristotle it was based on
a quite different principle. But it is sufficient that the circle should
be taken to be equal to any polygon that can be drawn intermediate
between the two ultimate polygons, and this gets over Proclus's difficulty. t
THE SQUARING OF THE CIRCLE 225
curvilinear figure to be measured, is particularly characteristic
of Archimedes.
We come now to the real rectifications or quadratures of
circles effected by means of higher curves, the construction
of which is more ' mechanical ' than that of the circle. Some
of these curves were applied to solve more than one of the
three classical problems, and it is not always easy to determine
for which purpose they were originally destined by their
inventors, because the accounts of the different authorities
do not quite agree. lamblichus, speaking of the quadrature
of the circle, said that
'Archimedes effected it by means of the spiral- shaped curve,
Nicomedes by means of the curve known by the special name
quadratrix (reTpayoavL^ova-a), Apollonius by means of a certain
curve which he himself calls " sister of the cochloid " but
which is the same as Nicomedes's curve, and finally Carpus
by means of a certain curve which he simply calls (the curve
arising) "from a double motion".' 1
Pappus says that
' for the squaring of the circle Dinostratus, Nicomedes and
certain other and later geometers used a certain curve which
took its name from its property ; for those geometers called it
quadratrix! 2
Lastly, Proclus, speaking of the trisection of any angle,
says that
' Nicomedes trisected any rectilineal angle by means of the
conchoidal curves, the construction, order and properties of
which he handed down, being himself the discoverer of their
peculiar character. Others have done the same thing by
means of the quadratrices of Hippias and Nicomedes. . . .
Others again, starting from the spirals of Archimedes, divided
any given rectilineal angle in any given ratio.' 3
All these passages refer to the quadratrix invented by
Hippias of Elis. The first two seem to imply that it was not
used by Hippias himself for squaring the circle, but that it
was Dinostratus (a brother of Menaechmus) and other later
geometers who first applied it to that purpose; lamblichus
and Pappus do not even mention the name of Hippias. We
might conclude that Hippias originally intended his curve to
1 Iambi, ap. Simpl. in Categ., p. 192. 19-24 K., 64 b 13-18 Br.
2 Pappus, iv, pp. 250. 33-252. 3. 3 Proclus on Eucl. I, p. 272. 1-12.
1523 Q
226 THE SQUARING OF THE CIRCLE
be used for trisecting an angle. But this becomes more doubt-
ful when the passages of Proclus are considered. Pappus's
authority seems to be Sporus, who was only slightly older
than Pappus himself (towards the end of the third century A.D.),
and who was the author of a compilation called K-qpia con-
taining, among other things, mathematical extracts on the
quadrature of the circle and the duplication of the cube.
Proclus's authority, on the other hand, is doubtless Geminus,
who was much earlier (first century B.C.) Now not only
does the above passage of Proclus make it possible that the
name quadratrix may have been used by Hippias himself,
but in another place Proclus (i.e. Geminus) says that different
mathematicians have explained the properties of particular
kinds of curves :
' thus Apollonius shows in the case of each of the conic curves
what is its property, and similarly Nicomedes with the
conchoids, Hippias %vith the quadratrices, and Perseus with
the spiric curves.' 1
This suggests that Geminus had before him a regular treatise
by Hippias on the properties of the quadratrix (which may
have disappeared by the time of Sporus), and that Nicomedes
did not write any such general work on that curve ; and,
if this is so, it seems not impossible that Hippias himself
discovered that it would serve to rectify, and therefore to
square, the circle.
(a) The Quadratrix of Hippias.
The method of constructing the curve is described by
c Pappus. 2 Suppose that A BCD is
a square, and BED a quadrant of a
circle with centre A.
Suppose (1) that a radius of the
circle moves uniformly about A from
the position AB to the position AD,
and (2) that in the same time the
line BC moves uniformly, always
A h mg d parallel to itself and with its ex-
tremity B moving along BA, from the position BC to the
position AD.
1 Proclus on Eucl. I, p. 356. 6-12. 2 Pappus, iv, pp. 252 sq.
THE QUADRATRIX OF HIPPIAS 227
Then, in their ultimate positions, the moving straight line
and the moving radius will both coincide with A D ; and at
any previous instant during the motion the moving line and
the moving radius will by their intersection determine a point,
as F or L.
The locus of these points is the quadratrix.
The property of the curve is that
LB AD- LEAD = (arc BED) : (arc ED) = AB-.FH.
In other words, if <p is the angle FAD made by any radius
vector AF with AD, p the length of AF, and a the length
of the side of the square,
p sin _ (p
a \ir
Now clearly, when the curve is once constructed, it enables
us not only to trisect the angle EAD but also to divide it in
any given ratio.
For let FH be divided at F r in the given ratio. Draw F'L
.parallel to AD to meet the curve in L : join AL, and produce
it to meet the circle in A".
Then the angles FAN, NAD are in the ratio of FF' to F'H,
as is easily proved.
Thus the quadratrix lends itself quite readily to the division
of any angle in a given ratio.
The application of the quadratrix to the rectification of the
circle is a more difficult matter, because it requires us to
know the position of G, the point where the quadratrix
intersects AD. This difficulty was fully appreciated in ancient
times, as we shall see.
Meantime, assuming that the quadratrix intersects AD
in G, we have to prove the proposition which gives the length
of the arc of the quadrant BED and therefore of the circum-
ference of the circle. This proposition is to the effect that
(arc of quadrant BED) :AB = AB:AG.
This is proved by reductio ad absurdura.
If the former ratio is not equal to AB:AG, it must be
equal to AB.AK, where AK is either (1) greater or (2) less
than AG.
(1) Let AK be greater than AG; and with A as centre
Q 2
228
THE SQUARING OF THE CIRCLE
and AK as radius, draw the quadrant KFL cutting the quad-
ratrix in F and AB in L.
Join AF, and produce it to meet the circumference BED
in E; draw FH perpendicular to AB.
Now, by hypothesis,
(arc BED) :AB = AB:AK
= (arc BED) .(arc LFK);
therefore A B = (arc LFK).
But, by the property of the quadra-
trix,
AB:FH = (arc BED) : (arc ED)
= (arc LFK): (arc FK);
and it was proved that AB — (arc LFK) ;
therefore FH = (arc FK) :
which is absurd. Therefore AK is not greater than AG.
(2) Let AK be less than .4£.
With centre ^1 and radius AK draw the quadrant KML.
Draw KF at right angles to AD meeting the quadratrix
in F; join AF, and let it meet the
quadrants in M, E respectively.
Then, as before, we prove that
AB = (arc LMK).
And, by the property of the quad-
ratriXy
AB : FK - (arc BED) : (arc DE)
= (arc LMK): (arc MK).
AB^ (arc LMK),
FK = (arc .Of):
which is absurd. Therefore J.i£ is not less than AG.
Since then AK is neither less nor greater than AG, it is
equal to it, and
(arc BE D):AB = AB: AG.
[The above proof is presumably due to Dinostratus (if not
to Hippias himself), and, as Dinostratus was a brother of
Menaechmus, a pupil of Eudoxus, and therefore probably
i K G
Therefore, since
THE QUADRATRIX OF HIPPIAS 229
flourished about 350 B.C., that is to say, some time before
Euclid, it is worth while to note certain propositions which
are assumed as known. These are, in addition to the theorem
of Eucl. VI. . c 3, the following: (1) the circumferences of
circles are as their respective radii; (2) any arc of a circle
is greater than the chord subtending it; (3) any arc of a
circle less than a quadrant is less than the portion of the
tangent at one extremity of the arc cut off by the radius
passing through the other extremity. (2) and (3) are of
course equivalent to the facts that, if oc be the circular measure
of an angle less than a right angle, sin oc < oc < tan a.]
Even now we have only rectified the circle. To square it
we have to use the proposition (1) in Archimedes's Measure-
ment of a Circle, to the effect that the area of a circle is equal
to that of a right-angled triangle in which the perpendicular
is equal to the radius, and the base to the circumference,
of the circle. This proposition is proved by the method of
exhaustion and may have been known to Dinostratus, who
was later than Eudoxus, if not to Hippias.
The criticisms of Sporus, 1 in which Pappus concurs, are
worth quoting :
(1) 'The very thing for which the construction is thought
to serve is actually assumed in the hypothesis. For how is it
possible, with two points starting from B, to make one of
them move along a straight line to A and the other along
a circumference to D in an equal time, unless you first know
the ratio of the straight line AB to the circumference BED ?
In fact this ratio must also be that of the speeds of motion.
For, if you employ speeds not definitely adjusted (to this
ratio), how can you make the motions end at the same
moment, unless this should sometime happen by pure chance ?
Is not the thing thus shown to be absurd 1
(2) 'Again, the extremity of the, curve which they employ
for squaring the circle, I mean the point in which the curve
cuts the straight line AD, is not found at all. For if, in the
figure, the straight lines OB, BA are made to end their motion
together, they will then coincide with AD itself and will not
cut one another any more. In fact they cease to intersect
before they coincide with AD, and yet it was the intersection
of these lines which was supposed to give the extremity of the
1 Pappus, iv, pp. 252. 26-254. 22.
230 THE SQUARING OF THE CIRCLE
curve, where it met the straight line AD. Unless indeed any
one should assert that the curve is conceived to be produced
further, in the same way as we suppose straight lines to be
produced, as far as AD. But this does not follow from the
assumptions made; the point G can only be found by first
assuming (as known) the ratio of the circumference to the
straight line.'
The second of these objections is undoubtedly sound. The
point G can in fact only be found by applying the method
of exhaustion in the orthodox Greek manner; e.g. we may
first bisect the angle of the quadrant, then the half towards
AD, then the half of that and so on, drawing each time
from the points F in which the bisectors cut the quadratrix
perpendiculars FH on AD and describing circles with AF
as radius cutting AD in K. Then, if we continue this process
long enough, HK will get smaller and smaller and, as G lies
between H and K, we can approximate to the position of G as
nearly as we please. But this process is the equivalent of
approximating to it, which is the very object of the whole
construction.
As regards objection (1) Hultsch has argued that it is not
valid because, with our modern facilities for making instru-
ments of precision, there is no difficulty in making the two
uniform motions take the same time. Thus an accurate clock
will show the minute hand describing an exact quadrant in
a definite time, and it is quite practicable now to contrive a
uniform rectilinear motion taking exactly the same time.
I suspect, however, that the rectilinear motion would be the
result of converting some one or more circular motions into
rectilinear motions ; if so, they would involve the use of an
approximate value of 7r, in which case the solution would depend
on the assumption of the very thing to be found. I am inclined,
therefore, to think that both Sporus's objections are valid.
(P) The Spiral of Archimedes.
We are assured that Archimedes actually used the spiral
for squaring the circle. He does in fact show how to rectify
a circle by means of a polar subtangent to the spiral. The
spiral is thus generated : suppose that a straight line with
one extremity fixed starts from a fixed position (the initial
THE SPIRAL OF ARCHIMEDES 231
line) and revolves uniformly about the fixed extremity, while
a point also moves uniformly along the moving straight line
starting from the fixed extremity (the origin) at the com-
mencement of the straight line's motion ; the curve described
is a spiral.
The polar equation of the curve is obviously p — a6.
Suppose that the tangent at any point P of the spiral is
met at T by a straight line dr%wn from 0, the origin or pole,
perpendicular to the radius vector OP; then OT is the polar
subtangent.
Now in the book On Spirals Archimedes proves generally
the equivalent of the fact that, if p be the radius vector to
the point P,
OT = p 2 /a.
If P is on the nth turn of the spiral, the moving straight
line will have moved through an angle 2(n — 1)tt + 6, say.
Hence p = a { 2 (n — l)ir + 6},
and 0T = p 2 /a = p{2(n~ \)tt + 6\.
Archiinedes's way of expressing this is to say (Prop. 20)
that, if p be the circumference of the circle with radius
0P(= p) } and if this circle cut the initial line in the point K,
OT — (n — l)p + arc KP measured ' forward ' from K to P.
If P is the end of the Tith turn, this reduces to
OT = n (circumf. of circle with radius OP),
and, if P is the end of the first turn in particular,
OT = (circumf. of circle with radius OP). (Prop, 19.)
The spiral can thus be used for the rectification of any
circle. And the quadrature follows directly from Measure-
ment of a Circle, Prop. 1.
(y) Solutions by Apollonius and Carpus.
Iamblichus says that Apollonius himself called the curve by
means of which he squared the circle ' sister of the cochloid '.
What this curve was is uncertain. As the passage goes on to
say that it was really ' the same as the (curve) of Nicomedes ',
and the quadratrix has just been mentioned as the curve used
232 THE SQUARING OF THE CIRCLE
by Nicomedes, some have supposed the ' sister of the cochloid '
(or conchoid) to be the quadratrix, but this seems highly im-
probable. There is, however, another possibility. Apollonius
is known to have written a regular treatise on the Cochliax,
which was the cylindrical helix. 1 It is conceivable that he
might call the cochlias the ' sister of the cochloid ' on the
ground of the similarity of the names, if not of the curves.
And, as a matter of fact, the drawing of a tangent to the
helix enables the circular section of the cylinder to be squared.
For, if a plane be drawn at right angles to the axis of the
cylinder through the initial position of the moving radius
which describes the helix, and if we project on this plane
the portion of the tangent at any point of the helix intercepted
between the point and the plane, the projection is equal to
an arc of the circular section of the cylinder subtended by an
angle at the centre equal to the angle through which the
plane through the axis and the moving radius has turned
from its original position. And this squaring by means of
what we may call the ' subtangent ' is sufficiently parallel to
the use by Archimedes of the polar subtangent to the spiral
for the same purpose to make the hypothesis attractive.
Nothing whatever is known of Carpus's curve ' of double
motion '. Tannery thought it was the cycloid ; but there is no
evidence for this.
(8) AiJproximations to the value of n.
As we have seen, Archimedes, by inscribing and cir-
cumscribing regular polygons of 96 sides, and calculating
their perimeters respectively, obtained the approximation
3i_ >7r >3io (Measurement of a Circle, Prop. 3). But we
now learn 2 that, in a work on Plinthides and Cylinders, he
made a nearer approximation still. Unfortunately the figures
as they stand in the Greek text are incorrect, the lower limit
tea *5"
being given as the ratio of u^cooe to \l£vuol, or 211875:67441
(= 3-141635), and the higher limit as the ratio of fx^co7rrj to
lifirva or 197888:62351 (= 3-17377), so that the lower limit
1 Pappus, viii, p. 1110. 20; Proclus on Eucl. I, p. 105. 5.
2 Heron, Metrica, i. 26, p. 66. 13-17.
APPROXIMATIONS TO THE VALUE OF II 233
as given is greater than the true value, and the higher limit is
greater than the earlier upper limit 3^ . Slight corrections by
Tannery (/^acoo/? for fx^oooe and /jl^cottP for ^X^ 177 )) S ive
better figures, namely
195882 211872
- > 7T >
62351 67441
or 3-1416016 > tt > 3-1415904....
T <?
Another suggestion 1 is to correct f*X v l jLa m ^° Pfopo and
H-X * 71 " 7 ] m t° P^wr], giving
195888 211875
62351 > ^ > 67444
or 3-141697... > tt > 3-141495 ....
If either suggestion represents the true reading, the mean
between the two limits gives the same remarkably close
approximation 31 4 1596.
Ptolemy 2 gives a value for the ratio of the circumference
of a circle to its diameter expressed thus in sexagesimal
8 30 ,
fractions, y n A, i.e. 3+ — + — Q or 3-1416. He observes
60 60
that this is almost exactly the mean between the Archimedean
limits 31 and 3^£. It is, however, more exact than this mean,
and Ptolemy no doubt obtained his value independently. He
had the basis of the calculation ready to hand in his Table
of Chords. This Table gives the lengths of the chords of
a circle subtended by arcs of \°, 1°, lj°, and so on by half
degrees. The chords are expressed in terms of 120th parts
of the length of the diameter. If one such part be denoted
by l p , the chord subtended by an arc of 1° is given by the
Table in terms of this unit and sexagesimal fractions of it
thus, l p 2' 50". Since an angle of 1° at the centre subtends
a side of the regular polygon of 360 sides inscribed in the
circle, the perimeter of this polygon is 360 times l p 2' 50 /r
or, since l p = 1 /120th of the diameter, the perimeter of the
polygon expressed in terms of the diameter is 3 times 1 2' 50",
that is 3 8' 30", which is Ptolemy's figure for it.
1 J. L. Heiben in Nordisk Tidsskrift for Filologi, 3 e Ser. xx. Fasc. 1-2.
2 Ptolemy, Syntaon's, vi. 7, p. 513. 1-5, Heib.
234 THE SQUARING OF THE CIRCLE
There is evidence of a still closer calculation than Ptolemy's
due to some Greek whose name we do not know. The Indian
mathematician Aryabhatta (born a.d. 476) says in his Lessons
in Calculation : %
'To 100 add 4; multiply the sum by 8; add 62000 more
and thus (we have), for a diameter of 2 myriads, the approxi-
mate length of the circumference of the circle ' ;
that is, he gives §§§o§ or 3-1416 as the value of it. But the
way in which he expresses it points indubitably to a Greek
source, ' for the Greeks alone of all peoples made the myriad
the unit of the second order ' (Rodet).
This brings us to the notice at the end of Eutocius's com-
mentary on the Measurement of a Circle of Archimedes, which
records 1 that other mathematicians made similar approxima-
tions, though it does not give their results.
' It is to be observed that Apollonius of Perga solved the
same problem in his 'flKVTOKiov (" means of quick delivery "),
using other numbers and making the approximation closer
[than that of Archimedes]. While Apollonius's figures seem
to be more accurate, they do not serve the purpose which
Archimedes had in view ; for, as we said, his object in this
book was to find an approximate figure suitable for use in
daily life. Hence we cannot regard as appropriate the censure
of Sporus of Nicaea, who seems to charge Archimedes with
having failed to determine with accuracy (the length of) the
straight line which is equal to the circumference of the circle,
to judge by the passage in his Keria where Sporus observes
that his own teacher, meaning Phil on of Gadara, reduced (the
matter) to more exact numerical expression than Archimedes
did, I mean in his \ and ^f ; in fact people seem, one after the
other, to have failed to appreciate Archimedes's object. They
have also used multiplications and divisions of myriads, a
method not easy to follow for any one who has not gone
through a course of Magnus's LogisticaJ
It is possible that, as Apollonius used myriads, ' second
myriads ', ' third myriads ', &c, as orders of integral numbers,
he may have worked with the fractions ? 2 > &c. ;
1 Archimedes, ed. Heib., vol. iii, pp. 258-9.
APPROXIMATIONS TO THE VALUE OF U 235
in any case Magnus (apparently later than Sporus, and therefore
perhaps belonging to the fourth or fifth century A. D.) would
seem to have written an exposition of such a method, which,
as Eutocius indicates, must have been very much more
troublesome than the method of sexagesimal fractions used
by Ptolemy.
The Trisection of any Angle.
This problem presumably arose from attempts to continue
the construction of regular polygons after that of the pentagon
had been discovered. The trisection of an angle would be
necessary in order to construct a regular polygon the sides
of which are nine, or any multiple of nine, in number.
A regular polygon of seven sides, on the other hand, would
no doubt be constructed with the help of the first discovered
method of dividing any angle in a given ratio, i.e. by means
of the quadratrix. This method covered the case of trisection,
but other more practicable ways of effecting this particular
construction were in due time evolved.
We are told that the ancients attempted, and failed, to
solve the problem by 'plane' methods, i.e. by means of the
straight line and circle ; they failed because the problem is
not ' plane ' but ' solid '. Moreover, they were not yet familiar
with conic sections, and so were at a loss; afterwards,
however, they succeeded in trisecting an angle by means of
conic sections, a method to which they were led by the
reduction of the problem to another, of the kind known as
vevo-tis (inclinationes , or vergings)}
(a) Reduction to a certain vevaLS, solved by conies.
The reduction is arrived at by the following analysis. It is
only necessary to deal with the case where the given angle to
be trisected is acute, since a right angle can be trisected
by drawing an equilateral triangle.
Let ABC be the given angle, and let AC be drawn perpen-
dicular to BC. Complete the parallelogram ACBF, and
produce the side FA to E.
1 Pappus, iv, p. 272. 7-14.
236
THE TRISECTION OF ANY ANGLE
Suppose E to be such a point that, if BE be joined meeting
AG in B, the intercept BE between AG and AE is equal
to 2AB.
Bisect BE at G. and join AG.
Then BG = GE = AG = AB.
Therefore lABG=AAGB=2AAEG
= 21 BBC, since FE, BC are parallel.
Hence
ADBC=±lABC,
and the angle A BG is trisected by BE.
Thus the problem is reduced to drawing BE from B to cut
AG and AE in such a %vay that the intercept BE = 2 AB.
In the phraseology of the problems called vevaeis the
problem is to insert a straight line EB of given length
2 A B between AE and A C in such a way that EB verges
towards B.
Pappus shows how to solve this problem in a more general
form. Given a parallelogram ABCB (which need not be
rectangular, as Pappus makes it), to draw AEF to meet GB
and BG produced in points E and F such that EF has a given
length.
Suppose the problem solved, EF being of the given length.
Complete the parallelogram
EDGE.
Then, EF being given in length,
BG is given in length.
Therefore G lies on a circle with
centre B and radius equal to the
given length.
Again, by the help of Eucl. I. 43 relating to the complements
REDUCTION TO A NETZIZ 237
of the parallelograms about the diagonal of the complete
parallelogram, we see that
BC.CD = BF.ED
= BF. FG.
Consequently G lies on a hyperbola with BF, BA as
asymptotes and passing through D.
Thus, in order to effect the construction, we have only to
draw this hyperbola as well as the circle with centre D and
radius equal to the given length. Their intersection gives the
point G, and E, i^are then determined by drawing GF parallel
to DC to meet BC produced in F and joining AF.
(/3) The vedai? equivalent to a cubic equation:
It is easily seen that the solution of the v evens is equivalent
to the solution of a cubic equation. For in the first figure on
p. 236, if FA be the axis of x, FB the axis of y, FA = a,
FB — b, the solution of the problem by means of conies as
Pappus gives it is the equivalent of finding a certain point
as the intersection of the conies
xy — ab,
(x - af + (y-b) 2 =4: (a 2 + b 2 ).
The second equation gives
(x + a)(x—3a) = (y + b) (3 b — y).
From the first equation it is easily seen that
(x + a) : (y + b) = a : y,
and that (x—3a)y = a(b—3y);
therefore, eliminating x, we have
a 2 (b-3y) = y 2 (3b-y),
or y*—3by 2 -3a 2 y + a 2 b = 0.
Now suppose that I ABC — 6, so that tan = b/a;
and suppose that t = tan DBC,
so that y = at.
We have then
a 3 t*-3ba 2 t 2 -3aH + a 2 b = 0.
238 THE TRISECTION OF ANY ANGLE
or at z — 3bt 2 —3at + b = 0,
whence b(l-3t 2 ) = a(3t-t 3 ),
, ' b 3t-t*
or tan 8 = - = - — —7.3
a l — 3t 2
so that, by the well-known trigonometrical formula,
t — tan J ;
that is, 5Z) trisects the angle ABC.
(y) T/ie Conchoids of Nicomedes.
Nicomedes invented a curve for the specific purpose of
solving such veixreis as the above. His date can be fixed with
sufficient accuracy by the facts (1) that he seems to have
criticized unfavourably Eratosthenes's solution of the problem
of the two mean proportionals or the duplication of the cube,
and (2) that Apollonius called a certain curve the 'sister of
the cochloid ', evidently out of compliment to Nicomedes.
Nicomedes must therefore have been about intermediate
between Eratosthenes (a little younger than Archimedes, and
therefore born about 280 B.C.) and Apollonius (born probably
about 264 B.C.).
The curve is called by Pappus the cochloid (Kox^oeiSijs
ypa/i/irj), and this was evidently the original name for it ;
later, e.g. by Proclus, it was called the conchoid (KoyxoetSr)?
ypafifirj). There were varieties of the cochloidal curves ;
Pappus speaks of the 'first', 'second', 'third' and 'fourth',
observing that the ' first ' was used for trisecting an angle and
duplicating the cube, while the others were useful for other
investigations. 1 It is the ' first ' which concerns us here.
Nicomedes constructed it by means of a mechanical device
which may be described thus. 2 AB is a ruler with a slot
in it parallel to its length, FE a second ruler fixed at right
angles to the first, with a peg C fixed in it. A third ruler
PC pointed at P has a slot in it parallel to its length which
fits the peg C. D is a fixed peg on PC in a straight line
with the slot, and D can move freely along the slot in AB.
If then the ruler PC moves so that the peg D describes the
1 Pappus, iv, p. 244. 18-20. 2 lb., pp. 242-4.
THE CONCHOIDS OF tflCOMEDES 239
length of the slot in A B on each side of F, the extremity P of
the ruler describes the curve which is called a conchoid or
cochloid.
Nicomedes called the straight line AB the ruler
(kolv&v), the fixed point G the pole (noXos), and the constant
length PD the distance (SidorTTjua).
The fundamental property of the curve, which in polar
coordinates would now be denoted by the equation
r = a + b sec 0,
is that, if any radius vector be drawn from C to the curve, as
CP, the length intercepted on the radius vector between the
curve and the straight line AB is constant. Thus any v evens
in which one of the two given lines (between which the
straight line of given length is to be placed) is a straight line
can be solved by means of the intersection of the other line
with a certain conchoid having as its pole the fixed point
to which the inserted straight line must verge (yeveiv). Pappus
tells us that in practice the conchoid was not always actually
drawn but that ' some ', for greater convenience, moved a ruler
about the fixed point until by trial the intercept was found to
be equal to the given length. 1
In the figure above (p. 236) showing the reduction of the
trisection of an angle to a v everts the conchoid to be used
would have B for its £>ole, AC for the ' ruler ' or base, a length
equal to 2 AB for its distance; and E would be found as the
intersection of the conchoid with FA produced.
Proclus says that Nicomedes gave the construction, the
order, and the properties of the conchoidal lines 2 ; but nothing
1 Pappus, iv, p. 246. 15. 2 Proclus on Eucl. I, p. 272. 3-7.
240
THE TRISECTION OF ANY ANGLE
of his treatise has come down to us except the construction
of the ' first ' conchoid, its fundamental property, and the fact
that the curve has the ruler or base as an asymptote in
each direction. The distinction, however, drawn by Pappus
between the ' first ', ' second ', ' third ' and ' fourth ' conchoids
may well have been taken from the original treatise, directly
or indirectly. We are not told the nature of the conchoids
other than the ' first ', but it is probable that they were three
other curves produced by varying the conditions in the figure.
Let a be the distance or fixed intercept between the curve and
the base, b the distance of the pole from the base. Then
clearly, if along each radius vector drawn through the pole
we measure a backwards from the base towards the pole,
we get a conchoidal figure on the side of the base towards
the pole. This curve takes three forms according as a is
greater than, equal to. or less than b. Each of them has
the base for asymptote, but in the first of the three cases
the curve has a loop as shown in the figure, in the second
case it has a cusp at the pole, in the third it has no double
point. The most probable hypothesis seems to be that the
other three cochloidal curves mentioned by Pappus are these
three varieties.
(8) Another reduction to a vtvcris (Archimedes).
A proposition leading to the reduction of the trisection
of an angle to another vevais is included in the collection of
Lemmas (Liber Assumptorum) which has come to us under
ARCHIMEDES'S SOLUTION (BY NET2IZ) 241
»
the name of Archimedes through the Arabic. Though the
Lemmas cannot have been written by Archimedes in their
present form, because his name is quoted in them more than
once, it is probable that some of them are of Archimedean
origin, and especially is this the case with Prop. 8, since the
uevo-L? suggested by it is of very much the same kind as those
the solution of which is assumed in the treatise On Spirals,
Props. 5-8. The proposition is as follows.
If A B be any chord of a circle with centre 0, and AB be
produced to C so that BC is
equal to the radius, and if CO
meet the circle in D, E, then the
arc AE will be equal to three
times the arc BD.
Draw the chord EF parallel
to AB, and join OB, OF.
Since BO = BO,
Z BOO = Z BOO.
Now IC0F=2Z.0EF,
= 2 Z BOO, by parallels,
= 2 Z BOO.
Therefore Z BOF =3 1 BOD,
and (arc BF) = (arc AE) = 3 (arc BD).
By means of this proposition we can reduce the trisection of
the arc AE to a vevais. For, in order to find an arc which is
one-third of the arc AE, we have only to draw through A
a straight line ABC meeting the circle again in B and E0
produced in C, and such that BC is equal to the radius of the
circle.
(e) Direct solutions by means of conies.
Pappus gives two solutions of the trisection problem in
which conies are applied directly without any preliminary
reduction of the problem to a vtvcris}
1. The analysis leading to the first method is as follows.
Let iC be a straight line, and B a point without it such
that, if BA, BC be joined, the angle BCA is double of the
angle BAG.
1 Pappus, iv, pp. 282-4.
1523 \\
242
THE TRISECTION OF ANY ANGLE
Draw BD perpendicular to AC, and cut off DE along DA
equal to DC. Join BE.
Then, since BE = BC,
lBEC = BCE.
But IBEC=ABAE+£EBA,
and, by hypothesis,
LBGA = 2 /.BAE.
Therefore Z BAE + Z EBA = 2 1 BAE;
therefore Z BAE = Z ^.Btf,
or AE = BE.
Divide AC at G so that AG = 2 GC, or 0£ = ±AC.
Also let Z# be made equal to ED, so that CD = ±CF.
It follows that GD = J (4C- C/F) = l^i? 7 .
Now
Also
Therefore
(Eucl, II. 6)
so that
BD 2 = BE 2 -ED 2
= BE 2 -EF 2 .
DA.AF=AE 2 -EF 2
= BE 2 -EF 2 .
BD 2 = DA.AF
= 3 AD . DG, from above,
£Z) 2 : 42) . DG = 3 : 1
= 3,i£ 2 :Z^ 2 .
Hence Z) lies on a hyperbola with AG as transverse axis
and with conjugate axis equal to V3 . AG.
Now suppose we are required
to trisect an arc AB of a circle
with centre 0.
Draw the chord AB, divide it
at C so that AC = 2 (75, and
construct the hyperbola which
has AC for transverse axis and
a straight line equal to Vs . AC for Conjugate axis.
Let the hyperbola meet the circular arc in P. Join PA,
PO, PB.
SOLUTIONS BY MEANS OF CONICS 243
Then, by the above proposition,
lPBA = 2/.PAB.
Therefore their doubles are equal,
or AP0A = 2/_P0B,
and OP accordingly trisects the arc APB and the angle AOB.
2. ' Some ', says Pappus, set out another solution not in-
volving recourse to a vevcris, as follows.
Let RPS be an arc of a circle which it is required to
trisect.
Suppose it done, and let the arc SP be one-third of the
arc SPR. P>
Join RP, SP.
Then the angle RSP is equal
to twice the angle SRP.
Let SE bisect the angle RSP, R x N s
meeting RP in E, and draw EX, PN perpendicular to RS.
Then Z ERS = Z ESR, so that RE = ES.
Therefore RX =■ XS, and X is given.
Again RS : SP = RE : EP = RX : XX]
therefore RS : RX = SP : NX.
But RS=2RX;
therefore SP=2XX.
It follows that P lies on a hyperbola with S as focus and XE
as directrix, and with eccentricity 2.
Hence, in order to trisect the arc, we have only to bisect RS
at X, draw XE at right angles to RS, and then draw a hyper-
bola with S as focus, XE as directrix, and 2 as the eccentricity.
The hyperbola is the same as that used in the first solution.
The passage of Pappus from which this solution is taken is
remarkable as being one of three passages in Greek mathe-
matical works still extant (two being in Pappus and one in
a fragment of Anthemius on burning mirrors) which refer to
the focus-and-directrix property of conies. The second passage
in Pappus comes under the heading of Lemmas to the Surface-
Loci of Euclid. 1 Pappus there gives a complete proof of the
1 Pappus, vii, pp. 1004-1114.
R 2
244 THE DUPLICATION OF THE CUBE
theorem that, if the distance of a point from a fixed point is
in a given ratio to its distance from a fixed line, the locus of
the point is a conic section which is an ellipse, a parabola,
or a hyperbola according as the given ratio is less than, equal
to, or greater than, unity. The importance of these passages
lies in the fact that the Lemma was required for the
understanding of Euclid's treatise. We can hardly avoid
the conclusion that the property was used by Euclid in his
Surface-Loci, but was assumed as well known. It was, there-
fore, probably taken from some treatise current in Euclid's
time, perhaps from Aristaeus's work on Solid Loci.
The Duplication of the Cube, or the problem
of the two mean proportionals.
(a) History of the problem.
In his commentary on Archimedes, On the Sphere and
Cylinder, II. 1, Eutocius has preserved for us a precious
collection of solutions of this famous problem. 1 One of the
solutions is that of Eratosthenes, a younger contemporary of
Archimedes, and it is introduced by what purports to be
a letter from Eratosthenes to Ptolemy. This was Ptolemy
Euergetes, who at the beginning of his reign (245 B.C.) per-
suaded Eratosthenes to come from Athens to Alexandria to be
tutor to his son (Philopator). The supposed letter gives the
tradition regarding the origin of the problem and the history of
its solution up to the time of Eratosthenes. Then, after some
remarks on its usefulness for practical purposes, the author
describes the construction by which Eratosthenes himself solved
it, giving the proof of it at some length and adding directions
for making the instrument by which the construction could
be effected in practice. Next he says that the mechanical
contrivance represented by Eratosthenes was, ' in the votive
monument ', actually of bronze, and was fastened on with lead
close under the crrecpdv^ of the pillar. There was, further,
on the pillar the proof in a condensed form, with one figure,
and, at the end, an epigram. The supposed letter of Eratos-
thenes is a forgery, but the author rendered a real service
1 Archimedes, ed. Heib., vol. hi, pp. 54. 26-106. 24.
HISTORY OF THE PROBLEM 245
by actually quoting the proof and the epigram, which are the
genuine work of Eratosthenes.
Our document begins with the story that an ancient tragic
poet had represented Minos as putting up a tomb to Glaucus
but being dissatisfied with its being only 100 feet each way;
Minos was then represented as saying that it must be made
double the size, by increasing each of the dimensions in that
ratio. Naturally the poet ' was thought to have made a mis-
* take '. Von Wilamowitz has shown that the verses which
Minos is made to say cannot have been from any play by
Aeschylus, Sophocles, or Euripides. They are the work of
some obscure poet, and the ignorance of mathematics shown
by him is the only reason why they became notorious and so
survived. The letter goes on to say that
'Geometers took up the question and sought to find out
how one could double a given solid while keeping the same
shape ; the problem took the name of " the duplication of the
cube " because they started from a cube and sought to double
it. For a long time all their efforts were vain ; then Hippo-
crates of Chios discovered for the first time that, if we can
devise a way of finding two mean proportionals in continued
proportion between two straight lines the greater of which
is double of the less, the cube will be doubled; that is, one
puzzle (airop-qfia) was turned by him into another not less
difficult. After a time, so goes the story, certain Delians, who
were commanded by the oracle to double a certain altar, fell
into the same quandary as before.'
At this point the versions of the story diverge somewhat.
The pseudo-Eratosthenes continues as follows :
' They therefore sent over to beg the geometers who were
with Plato in the Academy to find them the solution. The
latter applying themselves diligently to the- problem of finding-
two mean proportionals between two given straight lines,
Archytas of Taras is said to have found them by means of
a half cylinder, and Eudoxus by means of the so-called curved
lines ; but, as it turned out, all their solutions were theoretical,
and no one of them was able to give a practical construction
for ordinary use, save to a certain small extent Menaechmus,
and that with difficulty.'
Fortunately we have Eratosthenes's own version in a quota-
tion by Theon of Smyrna :
1 Eratosthenes in his work entitled Platonicus rekites that,
24S THE DUPLICATION OF THE CUBE
when the god proclaimed to the Delians by the oracle that, if
they would get rid of a plague, they should construct an altar
double of the existing one, their craftsmen fell into great
perplexity in their efforts to discover how a solid could be made
double of a (similar) solid ; they therefore went to ask Plato
about it, and he replied that the oracle meant, not that the god
wanted an altar of double the size, but that he wished, in
setting them the task, to shame the Greeks for their neglect
of mathematics and their contempt for geometry.' 1
Eratosthenes's version may well be true ; and there is no
doubt that the question was studied in the Academy, solutions
being attributed to Eudoxus, Menaechmus, and even (though
erroneously) to Plato himself. The description by the pseudo-
Eratosthenes of the three solutions by Archytas, Eudoxus and
Menaechmus is little more than a paraphrase of the lines about
them in the genuine epigram of Eratosthenes,
' Do not seek to do the difficult business of the cylinders of
Archytas, or to cut the cones in the triads of Menaechmus, or
to draw such a curved form of lines as is described by the
god-fearing Eudoxus.'
The different versions are reflected in Plutarch, who in one
place gives Plato's answer to the Delians in almost the same
words as Eratosthenes, 2 and in another place tells us that
Plato referred the Delians to Eudoxus and Helicon of Cyzicus
for a solution of the problem. 3
After Hippocrates had discovered that the duplication of
the cube was equivalent to finding two mean proportionals in
continued proportion between two given straight lines, the
problem seems to have been attacked in the latter form
exclusively. The various solutions will now be reproduced
in chronological order.
(ft) Archytas.
The solution of Archytas is the most remarkable of all,
especially when his date is considered (first half of fourth cen-
tury b. a), because it is not a construction in a plane but a bold
1 Theon of Smyrna, p. 2. 3-12.
2 Plutarch, De E apud Delphos, c. 6, 386 E.
3 De genio Socratis, c. 7, 579 c, n. •
ARCHYTAS 247
construction in three dimensions, determining a certain point
as the intersection of three surfaces of revolution, (1) a right
cone, (2) a cylinder, (3) a tore or anchor-ring with inner
diameter nil. The intersection of the two latter surfaces
gives (says Archytas) a certain curve (which is in fact a curve
of double curvature), and the point required is found as the
point in which the cone meets this curve.
Suppose that AC, AB are the two straight lines between
which two mean proportionals are to be found, and let A C be
made the diameter of a circle and AB & chord in it.
Draw a semicircle with AG as diameter, but in a plane at
right angles to the plane of the circle ABC, and imagine this
semicircle to revolve about a straight line through A per-
pendicular to the plane of ABC (thus describing half a tore
with inner diameter nil).
Next draw a right half-cylinder on the semicircle ABC as
base ; this will cut the surface of the half-fore in a certain
curve.
Lastly let CD, the tangent to the circle ABC at the point C,
meet AB produced in D; and suppose the triangle ADC to
revolve about AG as axis. This will generate the surface
of a right circular cone ; the point B will describe a semicircle
BQE at right angles to the plane of ABC and having its
diameter BE at right angles to AC; and the surface of the
cone will meet in some point P the curve which is the inter-
section of the half- cylinder and the half- fore.
248 THE DUPLICATION OF THE CUBE
Let APC be the corresponding position of the revolving
semicircle, and let AC meet the circumference ABC in M.
Drawing PM perpendicular to the plane of ABC, we see
that it must meet the circumference of the circle ABC because
P is on the cylinder which stands on ABC as base.
Let AP meet the circumference of the semicircle BQE in Q,
and let AC' meet its diameter in N. Join PC', QM, QN.
Then, since both semicircles are perpendicular to the plane
ABC, so is their line of intersection QN [EucL'XI. 19].
Therefore QN is perpendicular to BE.
Therefore QN 2 = BN . NE = AN . NM, [Eucl. III. 35]
so that the angle AQM is a right angle.
But the angle APC is also right :
therefore MQ is parallel to C'P.
It follows, by similar triangles, that
C'A:AP = AP:AM = AM:AQ;
that is, AC : AP = AP : AM = AM: AB,
and AB, AM, AP, AC are in continued proportion, so that
AM, AP are the two mean proportionals required.
In the language of analytical geometry, if AC is the axis
of x, a line through A perpendicular to AC in the plane of
A BC the axis of y, and a line through A parallel to PM the
axis of z, then P is determined as the intersection of the
surfaces
a 2
(1) x 2 + y 2 -\-z 2 = To x2 > (the cone)
(2) x 2 + y 2 — ax, (the cylinder)
(3) x 2 + y 2 + z 2 = a V(x 2 + y 2 ), (the tore)
where AC = a, AB = b.
From the first two equations we obtain
x 2 + y 2 + z 2 = (x 2 + y 2 ) 2 /b 2 ,
and from this and (3) we have
a _ V{x 2 + y 2 + z 2 ) _ V{x 2 + y 2 )
V(x* + y 2 + z 2 ) " V(x 2 + y 2 ) ~T~
or AC.AP = AP:AM=AM:AB.
ARCHYTAS. EUDOXUS 249
Compounding the ratios, we have
AC:AB = (AM:ABf;
therefore the cube of side A M is to the cube of side AB as AC
is to AB.
In the particular case where AG = 2AB, AM' ] = 2AB'\
and the cube is doubled.
(y) Eudoxus.
Eutocius had evidently seen some document purporting to
give Eudoxus's solution, but it is clear that it must have
been an erroneous version. The epigram of Eratosthenes
says that Eudoxus solved the problem by means of lines
of a 'curved or bent form' (KaimvXov elSos kv y pa finals) .
According to Eutocius, while Eudoxus said in his preface
that he had discovered a solution by means of ' curved lines ',
yet, when he came to the proof, he made no use of such
lines, and further he committed an obvious error in that he
treated a certain discrete proportion as if it were continuous. 1
It may be that, while Eudoxus made use of what was really
a curvilinear locus, he did not actually draw the whole curve
but only indicated a point or two upon it sufficient for his
purpose. This may explain the first part of Eutocius's remark,
but in any case we cannot believe the second part ; Eudoxus
was too accomplished a mathematician to make any confusion
between a discrete and a continuous proportion. Presumably
the mistake which Eutocius found was made by some one
who wrongly transcribed the original ; but it cannot be too
much regretted, because it caused Eutocius to omit the solution
altogether from his account.
Tannery 2 made an ingenious suggestion to the effect that
\Cudoxus's construction was really adapted from that of
Archytas by what is practically projection on the plane
of the circle A BG in Archytas's construction. It is not difficult
to represent the projection on that plane of the curve of
intersection between the cone and the tore, and, when this
curve is drawn in the plane ABC, its intersection with the
circle ABC itself gives the point M in Archytas's figure.
1 Archimedes, ed. Heib., vol. iii, p. 56. 4-8.
2 Tannery, Memoires scimtifiques, vol. i, pp. 53-61.
250
THE DUPLICATION OF THE CUBE
The projection on the plane ABC of the intersection between
the cone and the tore is seen, by means of their equations
(1) and (3) above, to be
x 2 = - V{x 2 + y 2 ),
or, in polar coordinates referred to A as origin and AC as axis,
b 2
P =
a cos 2 '
It is easy to find any number of points on the curve. Take
the circle ABC, and let AC the diameter and AB a chord
B M
be the two given straight lines between which two mean
proportionals have to be found.
With the above notation
AC = a, AB = b;
and, if BFbe drawn perpendicular to AC,
AB 2 = AF.AC,
or AF = b 2 /a,
Take any point G on BF and join AG.
Then, if IGAF=0, AG = AFsqc6.
With A as centre and AG as radius draw a circle meeting
AC in H, and draw HL at right angles to AC, meeting AG
produced in L.
EUDOXUS. MENAECHMUS 251
Then AL = AH sec 6 = AG sec 6 = AFsec 2 6.
b 2
That is, if p = AL, p — — sec 2 0,
' a
and L is a point on the curve.
Similarly any number of other points on the curve may be
found. If the curve meets the circle ABC in M, the length
AM is the same as that of AM in the figure of Archytas's
solution.
And AM is the first of the two mean proportionals between
AB and AC. The second (= AP in the figure of Archytas's
solution) is easily found from the relation AM 2 = AB . AP,
and the problem is solved.
It must be admitted that Tannery's suggestion as to
Eudoxus's method is attractive ; but of course it is only a con-
jecture. To my mind the objection to it is that it is too close
an adaptation of Archytas's ideas. Eudoxus was, it is true,
a pupil of Archytas, and there is a good deal of similarity
of character between Archytas's construction of the curve of
double curvature and Eudoxus's construction of the spherical
lemniscate by means of revolving concentric spheres ; but
Eudoxus was, I think, too original a mathematician to con-
tent himself with a mere adaptation of Archytas's method
of solution.
(S) Menaechmus,
Two solutions by Menaechmus of the problem of finding
two mean proportionals are described by Eutocius ; both find
a certain point as the intersection between two conies, in
the one case two parabolas, in the other a parabola and
a rectangular hyperbola. The solutions are referred to in
Eratosthenes's epigram : ' do not ', says Eratosthenes, ' cut the
cone in the triads of Menaechmus.' From the solutions
coupled with this remark it is inferred that Menaechmus
was the discoverer of the conic sections.
Menaechmus, brother of Dinostratus, who used the quadra-
trix to square the circle, was a pupil of Eudoxus and flourished
about the middle of the fourth century B. c. The most attrac-
tive form of the story about the geometer and the king who
wanted a short cut to geometry is told of Menaechmus and
252 THE DUPLICATION OF THE CUBE
Alexander : ' O king/ said Menaechmus, ' for travelling over
the country there are royal roads and roads for common
citizens, but in geometry there is one road for all.' 1 A similar
story is indeed told of Euclid and Ptolemy ; but there would
be a temptation to transfer such a story at a later date to
the more famous mathematician. Menaechmus was evidently
a considerable mathematician ; he is associated by Proclus with
Amyclas of Heraclea, a friend of Plato, and with Dinostratus
as having ' made the whole of geometry more perfect '. 2
Beyond, however, the fact that the discovery of the conic
sections is attributed to him, we have very few notices relating
to his work. He is mentioned along with Aristotle and
Callippus as a supporter of the theory of concentric spheres
invented by Eudoxus, but as postulating a larger number of
spheres. 3 We gather from Proclus that he wrote on the
technology of mathematics; he discussed for instance the
difference between the broader meaning of the word element
(in which any proposition leading to another may be said
to be an element of it) and the stricter meaning of something
simple and fundamental standing to consequences drawn from
it in the relation of a principle, which is capable of being
universally applied and enters into the proof of all manner
of propositions. 4 Again, he did not agree in the distinction
between theorems and problems, but would have it that they
were all problems, though directed to two different objects 5 ;
he also discussed the important question of the convertibility
of theorems and the conditions necessary to it. G
If x } y are two mean proportionals between straight
lines a, b,
that is, if a:x = x:y = y:b,
then clearly x 2 = ay, y 1 = bx, and xy = ab.
It is easy for us to recognize here the Cartesian equations
of two parabolas referred to a diameter and the tangent at its
extremity, and of a hyperbola referred to its asymptotes.
But Menaechmus appears to have had not only to recognize,
1 Stobaeus, Eclogae, ii. 31, 115 (vol. ii, p. 228. 30, Wachsmuth).
2 Proclus on Eucl. 1, p. 67. 9.
3 Theon of Smyrna, pp 201. 22-202. 2.
4 Proclus on Eucl. I, pp. 72. 23-73. 14. 5 lb., p. 78. 8-13.
15 lb., p. 254. 4-5.
MENAECHMUS AND CONICS 253
but to discover, the existence of curves having the properties
corresponding to the Cartesian equations. He discovered
them in plane sections of right circular cones, and it would
doubtless be the properties of the principal ordinates in
relation to the abscissae on the axes which he would arrive
at first. Though only the parabola and the hyperbola are
wanted for the particular problem, he would certainly not
fail to find the ellipse and its property as well. But in the
case of the hyperbola he needed the property of the curve
with reference to the asymptotes, represented by the equation
xy = ab ; he must therefore have discovered the existence of
the asymptotes, and must have proved the property, at all
events for the rectangular hyperbola. The original method
of discovery of the conies will occupy us later. In the mean-
time it is obvious that the use of any two of the curves
x 2 — ay, y 2 = bx, xy = ab gives the solution of our problem,
and it was in fact the intersection of the second and third
which Menaechmus used in his first solution, while for his
second solution he used the first two. Eutocius gives the
analysis and synthesis of each solution in full. I shall repro-
duce them as shortly as possible, only suppressing the use of
four separate lines representing the two given straight lines
and the two required means in the figure of the first solution.
First solution.
Suppose that AO, OB are two given straight lines of which
AO > OB, and let them form a right angle at 0.
Suppose the problem solved, and let the two mean propor-
tionals be OM measured along BO produced and ON measured
along AO produced. Complete the rectangle OMPN.
Then, since AO :0M = OM :0N = ON: OB,
we have (1) OB.OM = ON 2 = PM 2 ,
so that P lies on a parabola which has for vertex, OM for
axis, and OB for latus rectum ;
and (2) AO . OB = OM .ON = PN.PM,
so that P lies on a hyperbola with as centre and OM, OiYas
asymptotes.
254
THE DUPLICATION OF THE CUBE
Accordingly, to find the point P, we have to construct
(1) a parabola with as vertex, OM as axis, and latus rectum
equal to OB,
y
M
A a ^>-
fp
A o
N *
b
B
(2) a hyperbola with asymptotes OM, ON and such that
the rectangle contained by straight lines PM, PN drawn
from any point P on the curve parallel to one asymptote and
meeting the other is equal to the rectangle AO . OB.
The intersection of the parabola and hyperbola gives the
point P which solves the problem, for
AO-.PN = PN:PM = PM: OB.
Second solution.
Supposing the problem solved, as in the first case, we have,
since AO-.OM = OM: ON = ON: OB,
(1) the relation OB . OM = ON 2 = PM 2 ,
y
M
\
P
f^.
A
b
V r
4 x
MENAECHMUS AND CONICS 255
so that P lies on a parabola which has for vertex, OM for
axis, and OB for latus rectum,
(2) the similar relation AO . ON = OM 2 = PN 2 ,
so that P lies on a parabola which has for vertex, ON for
axis, and A for latus rectum.
In order therefore to find P, we have only to construct the
two parabolas with OM, ON for axes and OB, OA for latera
recta respectively ; the intersection of the two parabolas gives
a point P such that %
AO:PN=PN:PM=PM:OB,
and the problem is solved.
(We shall see later on that Menaechmus did not use the
names parabola and hyperbola to describe the curves, those
names being due to Apollonius.)
(e) The solution attributed to Plato.
This is the first in Eutocius's arrangement of the various
solutions reproduced by him. But there is almost conclusive
reason for thinking that it is wrongly attributed to Plato.
No one but Eutocius mentions it, and there is no reference to
it in Eratosthenes's epigram, whereas, if a solution by Plato
had then been known, it could hardly fail to have been
mentioned along with those of Archytas, Menaechmus, and
Eudoxus. Again, Plutarch says that Plato told the Delians
that the problem of the two mean proportionals was no easy
one, but that Eudoxus or Helicon of Cyzicus would solve it
for them ; he did not apparently propose to attack it himself.
And, lastly, the solution attributed to him is mechanical,
whereas we are twice told that Plato objected to mechanical
solutions as destroying the good of geometry. 1 Attempts
have been made to reconcile the contrary traditions. It is
argued that, while Plato objected to mechanical solutions on
principle, he wished to show how easy it was to discover
such solutions and put forward that attributed to him as an
illustration of the fact. I prefer to treat the silence of
Eratosthenes as conclusive on the point, and to suppose that
the solution was invented in the Academy by "some one con-
temporary with or later than Menaechmus.
1 Plutarch, Quaest. Conviv. 8. 2. 1, p. 718 e, f ; Vita Marcelli, c. 14. 5.
256
THE DUPLICATION OF THE CUBE
For, if we look at the figure of Menaechmus's second solu-
tion, we shall see that the given straight lines and the two
means between them are shown in cyclic order (clockwise)
as straight lines radiating from and separated by right
angles. This is exactly the arrangement of the lines in
1 Plato's ' solution. Hence it seems probable that some one
who had Menaechmus's second solution before him wished
to show how the same representation of the four straight
lines could be got by a mechanical construction as an alterna-
tive to the use of conies.
Drawing the two given straight lines with the means, that
is to say, OA, OM, ON, OB, in cyclic clockwise order, as in
Menaechmus's second solution, we have
AO:OM = OM :ON=ON: OB,
and it is clear that, if AM, MN, NB are joined, the angles
AMN, MNB are both right angles. The problem then is,
given OA, OB at right angles to one another, to contrive the
rest of the figure so that the angles at M, N are right.
The instrument used is somewhat like that which a shoe-
maker «uses to measure the length of the foot. FGH is a rigid
right angle made, say, of wood. KL is a strut which, fastened,
say, to a stick KF which slides along GF, can move while
remaining always parallel to GH or at right angles to GF.
Now place the rigid right angle FGH so that the leg GH
passes through B, and turn it until the angle G lies on iO
THE SOLUTION ATTRIBUTED TO PLATO 257
produced. Then slide the movable strut KL, which remains
always parallel to GH, until its edge (towards GH) passes
through A. If now the inner angular point between the
strut KL and the leg FG does not lie on BO produced,
the machine has to be turned again and the strut moved
until the said point does lie on BO produced, as M, care being
taken that during the whole of the motion the inner edges
of KL and HG pass through A, B respectively and the inner
angular point at G moves along AO produced.
That it is possible for the machine to take up the desired
position is clear from the figure of Menaechmus, in which
MO, NO are the means between AO and BO and the angles
AMN, MNB are right angles, although to get it into the
required position is perhaps not quite easy.
The matter may be looked at analytically thus. Let us
take any other position of the machine in which the strut and
the leg GH pass through A, B respectively, while G lies on AO
produced, but P, the angular point between the strut KL and
the leg FG, does not lie on OM produced. Take ON, OM as
the axes of x, y respectively. Draw PR perpendicular to OG,
and produce GP to meet OM produced in S.
Let
1523
AO = a, BO = b, 0G = r.
8
258 THE DUPLICATION OF THE CUBE
Then AR.RG = PR 2 ,
or (a + x) (r — x) = y 2 . (1)
Also, by similar triangles,
PR:RG = SO:OG
= OG:OB;
y r
or
(2)
r — x ~ b
From the equation (1) we obtain
x 2 + y 2 + ax
r = — — 3
a + x
and, by multiplying (1) and (2), we have
by (a + x) = ry 2 ,
whence, substituting the value of r, we obtain, as the locus of
P, a curve of the third degree,
b(a + x) 2 = y (x 2 + y 2 + ax).
The intersection (M) of this curve with the axis of y gives
0M d = a 2 b.
As a theoretical solution, therefore, 'Plato's' solution is
more difficult than that of Menaechmus.
({) Eratosthenes.
This is also a mechanical solution effected by means of
three plane figures (equal right-angled triangles or rectangles)
which can move parallel to one another and to their original
positions between two parallel rulers forming a sort of frame
and fitted with grooves so arranged that the figures can
move over one another. Pappus's account makes the figures
triangles, 1 Eutocius has parallelograms with diagonals drawn ;
triangles seem preferable. I shall use the lettering of Eutocius
for the second figure so far as it goes, but I shall use triangles
instead of rectangles.
1 Pappus, hi, pp. 56-8.
•
ERATOSTHENES 259
Suppose the frame bounded by the parallels AX, EY. The
m n a
initial position of the triangles is that shown in the first figure,
where the triangles are AMF, MNG, NQH.
In the second figure the straight lines AE, DH which are
m' m n' n
parallel to one another are those between which two mean
proportionals have to be found.
In the second figure the triangles (except AMF, which
remains fixed) are moved parallel to their original positions
towards AMF so that they overlap (as AMF, M'NG, N'QH),
NQH taking the position N'QH in which QH passes through D,
and MNG a position M'NG such that the points B, C where
MF, M'G and NG, N'H respectively intersect are in a straight
line with A, D.
Let AD, EH meet in K.
Then EK:KF=AK:KB
= FK:KG,
and EK : KF = AE : BF, while FK : KG = BF:CG;
therefore AE:BF=BF: CG.
Similarly BF : CG = CG : DH,
so that AE, BF, CG, DH are in continued proportion, and
BF, CG are the required mean proportionals.
This is substantially the short proof given in Eratosthenes's
s 2
260 THE DUPLICATION OF THE CUBE
inscription on the column; the construction was left to be
inferred from the single figure which corresponded to the
second above.
The epigram added by Eratosthenes was as follows :
' If. good friend, thou mindest to obtain from a small (cube)
a cube double of it, and duly to change any solid figure into
another, this is in thy power ; thou canst find the measure of
a fold, a pit, or the broad basin of a hollow well, by this
method, that is, if thou (thus) catch between two rulers (two)
means with their extreme ends converging. 1 Do not thou seek
to do the difficult business of Archytas's cylinders, or to cut the
cone in the triads of Menaechmus, or to compass such a curved
form of J lines as is described by the god-fearing Eudoxus.
Nay thou couldst, on these tablets, easily find a myriad of
means, beginning from a small base. Happy art thou,
Ptolemy, in that, as a father the equal of his son in youthful
vigour, thou hast thyself given him all that is dear to Muses
and Kings, and may he in the future, 2 O Zeus, god of heaven,
also receive the sceptre at thy hands. Thus may it be, and
let any one who sees this offering say "This is the gift of
Eratosthenes of Cyrene'V
(77) Nicomedes.
The solution by Nicomedes was contained in his book on
conchoids, and, according to Eutocius, he was inordinately
proud of it, claiming for it much superiority over the method
of Eratosthenes, which he derided as being impracticable as
well as ungeometrical.
Nicomedes reduced the problem to a v ever is which he solved
by means of the conchoid. Both Pappus and Eutocius explain
the method (the former twice over 3 ) with little variation.
Let AB, BC be the two straight lines between which two
means are to be found. Complete the parallelogram ABGL.
Bisect AB, BC in D and E.
Join LB, and produce it to meet GB produced in G.
Draw EF at right angles to BC and of such length that
GF = AD.
Join GF, and draw GH parallel to it.
1 Lit. 'converging with their extreme ends' (reppaaiv cixpois awBpo-
pudas).
2 Reading with v. Wilamowitz o $' «s varepov.
3 Pappus, iii, pp. 58. 23-62. 13; iv, pp. 246. 20-250. 25.
NICOMEDES
261
Then from the point F draw FHK cutting GH and EC
produced in H and K in such a way that the intercept
HK = GF= AD.
(This is done by means of a conchoid constructed with F as
pole, CH as ' ruler ', and < distance ' equal to AD or GF. This
conchoid meets EC produced in a point K. We then join FK
and, by the property of the conchoid, HK = the * distance '.)
Join KL, and produce it to meet BA produced in M.
Then shall CK, MA be the required mean proportionals.
For, since BG is bisected at E and produced to K,
BK.KC+CE 2 = EK 2 .
Add EF 2 to each ;
therefore BK . KC + CF 2 = KF 2 . (1)
Now, by parallels, MA-.AB = ML-.LK
= BC:CK.
But AB = 2 AD, and BG = \GC\
therefore MA : AD = GC : CK
= FH:HK,
and, componendo, MD : DA = FK : HK.
But, by construction, DA = HK ;
therefore MD = FK, and MD 2 = FK 2 .
262
THE DUPLICATION OF THE CUBE
Now MD 2 = BM . MA + DA 2 ,
while, by (1 ), FK 2 = BK. KC + Oi^ 2 ;
therefore 5ilf . TO + D A 2 = BK . KG + CF 2 .
But DA = CF; therefore BM.MA = BK . KG.
Therefore CZ : if .4 = BM.BK
= LC:CK;
while, at the same time, BM: BK = MA : AL.
Therefore LG : GK =CK:MA = MA:A L,
or AB : GK = GK:MA= MA : 50.
(0) Apollonius, Heron, Philon of Byzantium.
I give these solutions together because they really amount
the same thing. 1
Let AB, AC, placed at right angles, be the two given straight
lines. Complete the rectangle A BBC, and let E be the point
at which the diagonals bisect one another.
Then a circle with centre E and radius EB will circumscribe
the rectangle ABDC.
Now (Apollonius) draw with centre E a circle cutting
AB, AC produced in F, G but such that F, D, G are in one
straight line.
Or (Heron) place a ruler so that its edge passes through D,
1 Heron's solution is given in his Mechanics (i. 11) and Belopoeica, and is
reproduced by Pappus (iii, pp. 62-4) as well as by Eutocius (loc. cit).
APOLLONIUS, HERON, PHILON OF BYZANTIUM 263
and move it about D until the edge intersects AB, AC pro-
duced in points (F, G) which are equidistant from E.
Or (Philon) place a ruler so that it passes through D and
turn it round D until it cuts AB, AG produced and the circle
about ABDG in points F, G, H such that the intercepts FD,
HG are equal.
Clearly all three constructions give the same points F, G.
For in Pinion's construction, since FD = HG, the perpendicular
from E on DH, which bisects DH, must also bisect FG, so
that EF = EG.
We have first to prove that AF.FB = AG. GO.
(a) With Apollonius's and Heron's constructions we have, if
K be the middle point of AB,
AF.FB + BK 2 = FK 2 .
Add KE 2 to both sides ;
therefore AF.FB + BE 2 = EF 2 .
Similarly AG.GC+ CE 2 = EG 2 .
But BE = CE, and EF=EG;
therefore AF.FB = AG. GC.
(b) With Philon's construction, since GH — FD,
HF.FD = DG.GH.
But, since the circle BDHC passes through A,
HF. FD = AF. FB, and DG .GH = AG .GC;
therefore AF.FB = AG.GC.
Therefore FA:AG = CG:FB.
But, by similar triangles,
FA:AG = DC:CG, and also =FB:BD;
therefore DC : CG = CG :FB = FB: BD,
or AB : CG = CG:FB = FB: AC.
The connexion between this solution and that of Menaech-
mus can be seen thus. We saw that, if a : x = x : y = y : b,
x 2 = ay, y 2 — bx, xy = ah,
which equations represent, in Cartesian coordinates, two
parabolas and a hyperbola. Menaechmus in effect solved the
264 THE DUPLICATION OF THE CUBE
problem of the two mean proportionals by means of the points
of intersection of any two of these conies.
But, if we add the first two equations, we have
x 2 + y 2 — bx — ay = 0,
which is a circle passing through the points common to the
two parabolas x 2 = ay, y 2 = bx.
Therefore we can equally obtain a solution by means of
the intersections of the circle x 2 + y 2 — bx — ay = and the
rectangular hyperbola xy = ab.
This is in effect what Philon does, for, if AF, AG are the
coordinate axes, the circle x 2 -\-y 2 — bx — ay = is the circle
BDHC, and xy = ab is the rectangular hyperbola with
AF, AG as asymptotes and passing through D, which
hyperbola intersects the circle again in H, a point such
that FD = HG.
(l) Diodes and the cissoid.
We gather from allusions to the cissoid in Proclus's com-
mentary on Eucl. I that the curve which Geminus called by
that name was none other than the curve invented by Diocles
and used by him for doubling the cube or finding two mean
proportionals. Hence Diocles must have preceded Geminus
(fl. 70 B.C.). Again, we conclude from the two fragments
preserved by Eutocius of a work by him, irepl wvpeiodv, On
burning-mirrors, that he was later than Archimedes and
Apollonius. He may therefore have flourished towards the
end of the second century or at the beginning of the first
century B.C. Of the two fragments given by Eutocius one
contains a solution by means of conies of the problem of
dividing a sphere by a plane in such a way that the volumes
of the resulting segments shall be in a given ratio — a problem
equivalent to the solution of a certain cubic equation — while
the other gives the solution of the problem of the two mean
proportionals by means of the cissoid.
Suppose that AB, DG are diameters of a circle at right
angles to one another. Let E, F be points on the quadrants
BD, BG respectively such that the arcs BE, BF are equal.
Draw EG, FH perpendicular to DG. Join GE, and let P be
the point in which GE, FH intersect.
DIOCLES AND THE CISSQID
265
The cissoid is the locus of all the points P corresponding to
different positions of E on the quadrant BD and of F at an
equal distance from B on the quadrant BG.
If P is any point found by the above construction, it is
required to prove that FH, HG are two mean proportionals in
continued proportion between DH and HP, or that
DH:HF=HF:HG = HG:HP.
Now it is clear from the construction that EG — FH,
DG = HC, so that CG:GE = DH:HF.
And, since FH is a mean proportional between DH, HG,
DH:HF=HF:GH
But, by similar triangles,
CG:GE=GH:HP.
It follows that
DH : HF = HF: CH = CH: HP,
or FH, HG are the two mean proportionals between DH, HP.
[Since DH . HP — HF. GH, we have, if a is the radius of
the circle and if OH = x y HP = y, or (in other words) if we
use OG, OB as axes of coordinates,
(a + x) y = V (a 2 — x 2 ) .(a — x)
or y 2 (a + x) = (a — x) 3 ,
which is the Cartesian equation of the curve. It has a cusp
at G, and the tangent to the circle at D is an asymptote to it.]
266 THE DUPLICATION OF THE CUBE
Suppose now that the cissoid has been drawn as shown by
the dotted line in the figure, and that we are required to find
two mean proportionals between two straight lines a, b.
Take the point K on OB such that DO:OK=a:b.
Join DK, and produce it to meet, the cissoid in Q.
Through Q draw the ordinate LM perpendicular to DO.
Then, by the property of the cissoid, LM, MO are the two
mean proportionals between DM, MQ. And
DM:MQ = DO:OK = a:b.
In order, then, to obtain the two mean proportionals between
a and b, we have only to take straight lines which bear respec-
tively the same ratio to DM, LM, MO, MQ as a bears to DM.
The extremes are then a, b, and the two mean proportionals
are found.
(k) Sporus and Pappus.
The solutions of Sporus and Pappus are really the same as
that of Diocles, the only difference being that, instead of using
the cissoid, they use a ruler which they turn about a certain
point until certain intercepts which it cuts off between two
pairs of lines are equal.
In order to show the identity of the solutions, I shall draw
Sporus's figure with the same lettering as above for corre-
sponding points, and I shall add dotted lines to show the
additional auxiliary lines used by Pappus. 1 (Compared with
my figure, Sporus's is the other way up, and so is Pappus's
where it occurs in his own Synagoge, though not in Eutocius.)
Sporus was known to Pappus, as we have gathered from
Pappus's reference to his criticisms on the quadratrix, and
it is not unlikely that Sporus was either Pappus's master or
a fellow-student of his. But when Pappus gives (though in
better form, if we may judge by Eutocius's reproduction of
Sporus) the same solution as that of Sporus, and calls it
a solution kolO* 77/xay, he clearly means 'according to my
method ', not ' our method ', and it appears therefore that he
claimed the credit of it for himself.
Sporus makes DO, OK (at right angles to one another) the
actual given straight lines; Pappus, like Diocles, only takes
1 Pappus, iii, pp. 64-8 ; viii, pp. 1070-2.
SPORUS AND PAPPUS
267
them in the same proportion as the given straight lines.
Otherwise the construction is the game.
A circle being drawn with centre and radius DO, we join
DK and produce it to meet the circle in /.
Now conceive a ruler to pass through G and to be turned
about G until it cuts DI, OB and the circumference of the
circle in points Q, T, R such that QT = TR. Draw QM, RN
perpendicular to BG.
Then, since QT = TR, MO = ON, and MQ, 3¥R are equi-
distant from OB. Therefore in reality Q lies on the cissoid of
Diodes, and, as in the first part of Diocles's proof, we prove
(since RJS r is equal to the ordinate through Q, the foot of
which is M) that
DM : RN = RN: MG = MG : MQ,
and we have the two means between DM, MQ, so that we can
easily construct the two means between DO, OK.
But Sporus actually proves that the first of the two means
between DO and OK is T. This is obvious from the above
relations, because
RN: OT = GN\ GO = DM:DO = MQ: OK.
Sporus has an ab initio proof of the fact, but it is rather
confused, and Pappus's proof is better worth giving, especially
as it includes the actual duplication of the cube.
It is required to prove that DO : OK = DO 3 : 0T\
268 THE DUPLICATION OF THE CUBE
Join RO, and produce it to meet the circle at #. Join
DS, SO.
Then, since RO = OS and RT = TQ, SQ is parallel to AB
and meets OC in M.
Now
DM:MC= SM 2 : MC 2 = CM 2 : ifQ 2 (since Z RCS is right).
Multiply by the ratio CM : MQ ;
therefore (DM : MC) . (CM : MQ) = (CM 2 : MQ 2 ) . (CM : MQ)
or DM:MQ = CM S :MQ*.
But DM:MQ = DO:OK,
and CM:MQ = CO:OT.
Therefore DO : 0# = CO 3 : OT 3 = DO 3 : OT 3 .
Therefore OTis the first of the two mean proportionals to
DO, OK ; the second is found by taking a third proportional
to DO, OT.
And a cube has been increased in any given ratio.
(X) Apiiroximation to a solution by plane "methods only.
There remains the procedure described by Pappus and
criticized by him at length at the beginning of Book III of
his Collection} It was suggested by some one ' who was
thought to be a great geometer ', but whose name is not given.
Pappus maintains that the author did not understand what
he was about, ' for he claimed that he was in possession of
a method of finding two mean proportionals between two
straight lines by means of plane considerations only ' ; he
gave his construction to Pappus to examine and pronounce
upon, while Hierius the philosopher and other friends of his
supported his request for Pappus's opinion. The construction
is as follows.
Let the given straight lines be AB, AD placed at right
angles to one another, AB being the greater.
Draw BC parallel to AD and equal to AB. Join CD meeting
BA produced in E. Produce BC to L, and draw EL' through
E parallel to BL. Along CL cut off lengths CF, FG, GK, KL,
1 Pappus, iii, pp. 30-48.
APPROXIMATION BY PLANE METHODS 269
each of which is equal to BG. Draw GG', FF' , GG', KK', LI!
parallel to BA.
On LI/, KK' take LM, KB equal to BA, and bisect LM
in N.
Take P, Q on LL' such that L'L, L'N, L'P, L'Q are in con-
tinued proportion ; join QB, BL, and through iV draw NS
parallel to QB meeting BL in S.
Draw ST parallel to BL meeting GG' in T.
To G'G, G'T take continued proportionals G'O, G'U, as bef ore-
Take W on FF' such that FW ^ BA, join UW, WG, and
through T draw TI parallel to UW meeting WG in /.
Through i" draw 1 V parallel to BG meeting GG ' in F.
Take continued proportionals (7'C, G'V, G'X, G'Y, and draw
XZ, VZ f parallel to FD meeting EG in ^, ^ r . Lastly draw
ZX', Z'Y' parallel to BG.
Then, says the author, it is required to prove that ZX', Z'Y'
are two 'mean proportionals in continued proportion between
AD, BG.
Now, as Pappus noticed, the supposed conclusion is clearly
not true unless BY is parallel to BG, which in general it is not.
But what Pappus failed to observe is that, if the operation of
taking the continued proportionals as described is repeated,
not three times, but an infinite number of times, the length of
the line G'Y tends continually towards equality with EA.
Although, therefore, by continuing the construction we can
never exactly determine the required means, the method gives
an endless series of approximations tending towards the true
lengths of the means.
270 THE DUPLICATION OF THE CUBE
Let LL'=BE = a, AB = b, L'N=a (for there is no
necessity to take iV at the middle point of LM ).
Then L'Q = oi 6 /a 2 ,
therefore LQ = (a 3 — a 3 )/a 2 .
Ad TG SL NL_ (a-oc)a 2
RK~RL~ QL~ a 6 -oi 6 '
therefore TG = la -" )a * h ,
cr — a 05
and accordingly G'T = a- ^T^f 6 .
Now let a w be the length corresponding to G'T after n
operations ; then it is clear that
(a — <x n ) a 2 b
a — a
a*~*n
. oi n must approach some finite limit when n = go . Taking £
as this limit, we have
■ (a-g)a 2 b
and, £ = tt not being -a root of this equation, we get at once
£ 3 = o? — a 2 b = a 2 (a — b).
Therefore, ultimately C'V is one of the mean proportionals
between EA and EB, whence Y'Z' will be one of the mean
proportionals between AD, BC, that is, between AB and AB.
The above was pointed out for the first time by R. Pendle-
bury, 1 and I have followed his way of stating the matter.
1 Messenger of Mathematics, ser. 2, vol. ii (1873), pp. 166-8.
VIII
ZENO OF ELEA
We have already seen how the consideration of the subject
of infinitesimals was forced upon the Greek mathematicians so
soon as they came to close grips with the problem of the
quadrature of the circle. Antiphon the Sophist was the first
to indicate the correct road upon which the solution was to
be found, though he expressed his idea in a crude form which
was bound to provoke immediate and strong criticism from
logical minds. Antiphon had inscribed a series of successive
regular polygons in a circle, each of which had double as
many sides as the preceding, and he asserted that, by con-
tinuing this process, we should at length exhaust the circle :
'he thought that in this way the area of the circle would
sometime be used up and a polygon would be inscribed in the
circle the sides of which on account of their smallness would
coincide with the circumference.' 1 Aristotle roundly said that
this was a fallacy which it was not even necessary for a
geometer to trouble to refute, -since an expert in any science
is not called upon to refute all fallacies, but only those which
are false deductions from the admitted principles of the
science ; if the fallacy is based on anything which is in con-
tradiction to any of those principles, it may at once be ignored. 2
Evidently therefore, in Aristotle's view, Antiphon's argument
violated some 'geometrical principle', whether this was the
truth that a straight line, however short, can never coincide
with an arc of a circle, or the principle assumed by geometers
that geometrical magnitudes can be divided ad infinitum.
But Aristotle is only a representative of the criticisms
directed against the ideas implied in Antiphon's argument;
those ideas had already, as early as the time of Antiphon
1 Simpl. in Artist. Phys., p. 55. 6 Diels.
2 Arist. Phys. i. 2, 185 a 14-17.
272 ZENO OF ELEA
himself (a contemporary of Socrates), been subjected to a
destructive criticism expressed with unsurpassable piquancy
and force. No wonder that the subsequent course of Greek
geometry was profoundly affected by the arguments of Zeno
on motion. Aristotle indeed called them ' fallacies ', without
being able to refute them. The mathematicians, however, knew
better, and, realizing that Zeno's arguments were fatal to
infinitesimals, they saw that they could only avoid the diffi-
culties connected with them by once for all banishing the idea
of the infinite, even the potentially infinite, altogether from
their science ; thenceforth, therefore, they made no use of
magnitudes increasing or diminishing ad infinitum, but con-
tented themselves with finite magnitudes that can be made as
great or as small as we please,} If they used infinitesimals
at all, it was only as a tentative means of discovering proposi-
tions ; they proved them afterwards by rigorous geometrical
methods. An illustration of this is furnished by the Method of
Archimedes. In that treatise Archimedes finds (a) the areas
of curves, and (6) the volumes of solids, by treating them
respectively as the sums of an infinite number (a) of parallel
lines, i.e. infinitely narrow strips, and (b) of parallel planes,
i. e. infinitely thin laminae ; but he plainly declares that this
method is only useful for discovering results and does not
furnish a proof of them, but that to establish them scientific-
ally a geometrical proof by the method of exhaustion, with
its double reductio ad absurdum, is still necessary.
Notwithstanding that the criticisms of Zeno had so impor-
tant an influence upon the lines of development of Greek
geometry, it does not appear that Zeno himself was really
a mathematician or even a physicist. Plato mentions a work
of his (to. tov Zrjvtovos ypd/x/iara, or to arvyypafjifjia) in terms
which imply that it was his only known work. 2 Simplicius
too knows only one work of his, and this the same as that
mentioned by Plato 3 ; when Suidas mentions four, a Commen-
tary on or Exposition of Empedocles, Controversies, Against
the philosophers and On Nature, it may be that the last three
titles are only different designations for the one work, while
the book on Empedocles may have been wrongly attributed
1 Cf. Arist. Fhys. iii. 7, 207 b 31. 2 Plato, Purmenides, 127 c sq.
3 Simpl. in Phys., pp. 139. 5, HO. 27 Diels.
ZENO OF ELEA 273
to Zeno, 1 Plato puts into the mouth of Zeno himself an
explanation of the character and object of his book. 2 It was
a youthful effort, and it was stolen by some one, so that the
author had no opportunity of considering whether to publish
it or not. Its object was to defend the system of Parmenides
by attacking the common conceptions of things. Parmenides
held that only the One exists; whereupon common sense
pointed out that many contradictions and absurdities will
follow if this be admitted. Zeno replied that, if the popular
view that Many exist be accepted, still more absurd results
will follow. The work was divided into several parts (Xoyot
according to Plato) and each of these again into sections
('hypotheses' in Plato, 'contentions', kirL^LprjixaTa^ in Sim-
plicius) : each of the latter (which according to Proclus
numbered forty in all 3 ) seems to have taken one of the
assumptions made on the ordinary view of life and to have
shown that it leads to an absurdity. It is doubtless on
account of this systematic use of indirect proof by the reductio
ad absurdum of particular hypotheses that Zeno is said to
have been called by Aristotle the discoverer of Dialectic 4 ;
Plato, too, says of him that he understood how to make one
and the same thing appear like and unlike, one and many, at
rest and in motion. 5
Zeno's arguments about motion.
It does not appear that the full significance and value of
Zeno's paradoxes have ever been realized until these latter
days. The most modern view of them shall be expressed in
the writer's own words :
' In this capricious world nothing is more capricious than
posthumous fame. One of the most notable victims of pos-
terity's lack of judgement is the Eleatic Zeno. Having
invented four arguments all immeasurably subtle and pro-
found, the grossness of subsequent philosophers pronounced
him to be a mere ingenious juggler, and his arguments to be
1 Zeller, i 5 , p. 587 note.
2 Plato, Parmenides 128 C-E.
3 Proclus in Parm., p. 694. 23 seq.
4 Diog. L. viii. 57, ix. 25 ; Sext. Emp. Math. vii. 6.
5 Plato, Phaedrus 261 d.
1523 T
274 ZENO OF ELEA
one and all sophisms. After two thousand years of continual
refutation, these* sophisms were reinstated, and made the
foundation of a mathematical renaissance, by a German
professor who probably never dreamed of any connexion
between himself and Zeno. Weierstrass, by strictly banishing
all infinitesimals, has at last shown that we live in an
unchanging world, and that the arrow, at every moment of its
flight, is truly at rest. The only point where Zeno probably
erred was in inferring (if he did infer) that, because there
is no change, the world must be in the same state at one time
as at another. This consequence by no means follows, and in
this point the German professor is more constructive than the
ingenious Greek. Weierstrass, being able to embody his
opinions in mathematics, where familiarity with truth elimi-
nates the vulgar prejudices of common sense, has been able to
give to his propositions the respectable air of platitudes ; and
if the result is less delightful to the lover of reason than Zeno's
bold defiance, it is at any rate more calculated to appease the
mass of academic mankind.' x
Thus, while in the past the arguments of Zeno have been
treated with more or less disrespect as mere sophisms, we have
now come to the other extreme. It appears to be implied that
Zeno anticipated Weierstrass. This, I think, a calmer judge-
ment must pronounce to be incredible. If the arguments of
Zeno are found to be ' immeasurably subtle and profound '
because they contain ideas which Weierstrass used to create
a great mathematical theory, it does not follow that for Zeno
they meant at all the same thing as for Weierstrass. On the
contrary, it is probable that Zeno happened upon these ideas
without realizing any of the significance which Weierstrass
was destined to give them ; nor shall we give Zeno any less
credit on this account.
It is time to come to the arguments themselves. It is the
four arguments on the subject of motion which are most
important from the point of view of the mathematician ; but
they have points of contact with the arguments which Zeno
used to prove the non-existence of Many, in refutation of
those who attacked Parmenides's doctrine of the One. Accord-
ing to Simplicius, he showed that, if Many exist, they must
1 Bertiand Russell, The Principles of Mathematics, vol. i, 1903, pp.
347, 348.
ZENO'S ARGUMENTS ABOUT MOTION 275
be both great and small, so great on the one hand as to be
infinite in size and so small on the other as to have no size. 1
To prove the latter of these contentions, Zeno relied on the
infinite divisibility of bodies as evident ; assuming this, he
easily proved that division will continually give smaller and
smaller parts, there will be no limit to the diminution, and, if
there is a final element, it must be absolutely nothing. Conse-
quently to add any number of these m^-elements to anything
will not increase its size, nor will the subtraction of them
diminish it ; and of course to add them to one another, even
in infinite number, will give nothing as the total. (The
second horn of the dilemma, not apparently stated by Zeno
in this form, would be this. A critic might argue that infinite
division would only lead to parts having some size, so that the
last element would itself have some size ; to this the answer
would be that, as there would, by hypothesis, be an infinite
number of such parts, the original magnitude which was
divided would be infinite in size.) The connexion between
the arguments against the Many and those against motion
lies in the fact that the former rest on the assumption of
the divisibility of matter ad infinitum, and that this is the
hypothesis assumed in the first two arguments against motion.
We shall see that, while the first two arguments proceed on
this hypothesis, the last two appear to proceed on the opposite
hypothesis that space and time are not infinitely divisible, but
that they are composed of indivisible elements ; so that the
four arguments form a complete dilemma.
The four arguments against motion shall be stated in the
words of Aristotle.
I. The Dichotomy.
' There is no motion because that which is moved must
arrive at the middle (of its course) before it arrives at the
end.' 2 (And of course it must traverse the half of the half
before it reaches the middle, and so on ad infinitum.)
II. The Achilles.
'This asserts that the slower when running will never be
1 Simpl. in Phys., p. 139. 5, Diels.
2 Aristotle, Phys. vi. 9, 239 b 11.
T 2
276 ZENO OF ELEA
overtaken by the quicker; for that which is pursuing must
first reach the point from which that which is fleeing started,
so that the slower must necessarily always be some distance
ahead.' J
III. The Arrow.
' If, says Zeno, everything is either at rest or moving when
it occupies a space equal (to itself), while the object moved is
always in the instant (eVn 8' del to $tpb\itvov kv rS> vvv, in
the now), the moving arrow is unmoved.' 2
I agree in Brochard's interpretation of this passage, 3 from
which Zeller 4 would banish rj Kivdrou, ' or is moved '. The
argument is this. It is strictly impossible that the arrow can
move in the instant, supposed indivisible, for, if it changed its
position, the instant would be at once divided. Now the
moving object is, in the instant, either at rest or in motion ;
but, as it is not in motion, it is at rest, and as, by hypothesis,
time is composed of nothing but instants, the moving object is
always at rest. This interpretation has the advantage of
agreeing with that of Simplicius, 5 which seems preferable
to that of Themistius 6 on which Zeller relies.
IV. The Stadium. I translate the first two sentences of
Aristotle's account 7 :
{ The fourth is the argument concerning the two rows of
bodies each composed of an equal number of bodies of equal
size, which pass one another on a race-course as they proceed
with equal velocity in opposite directions, one row starting
from the end of the course and the other from the middle.
This, he thinks, involves the conclusion that half a given time
is equal to its double. The fallacy of the reasoning lies in
the assumption that an equal magnitude occupies an equal
time in passing with equal velocity a magnitude that is in
motion and a magnitude that is at rest, an assumption which
is false.'
Then follows a description of the process by means of
1 Aristotle, Phys. vi. 9, 239 b 14. 2 lb. 239 b 5-7.
3 V. Brochard, Etudes de Philosophie ancienne et de Philosophic modeme,
Paris 1912, p. 6.
4 Zeller, i 5 , p. 599. 5 Simpl. in Phys., pp. 1011-12, Diels.
6 Them, (ad loc, p. 392 Sp., p. 199 Sch.)
7 Phys. vi, 9, 239 b 33-240 a 18.
ZENO'S ARGUMENTS ABOUT MOTION 277
letters A, B, C the exact interpretation of which is a matter
of some doubt l ; the essence of it, however, is clear. The first
diagram below shows the original positions of the rows of
A i
A 2
A 3
A 4
*S.
A *
A r
A*
B 8
B,
B *
B 5
ii
B a
h
B,
E
jEE
bodies (say eight in number). The A's represent a row which
is stationary, the JS's and C"s are rows which move with equal
velocity alongside the A's and one another, in the directions
shown by the arrows. Then clearly there will be (1) a moment
^
A 2
A 3
A 4
A 5
A 6
A J
A 8
B 8
B y
B 6
B 5
B 4
Bs
B 2
Bi
Ci
C 2
C 3
^
c 5
c*
c r
c *
when the 5's and C's will be exactly under the respective A%
as in the second diagram, and after that (2) a moment when
the B's and C's will have exactly reversed their positions
relatively to the A's, as in the third figure.
A 1
A 2
A 3
A 4
A 5
\
A t\
A 8
B 8
By
B 6
B 5
B 4
B 3
Bj,
8 1
^l| C 2l^3l^4 £s_ ^G C y C&
The observation has been made 2 that the four arguments
form a system curiously symmetrical. The first and fourth
consider the continuous and movement within given limits,
the second ■ and third the continuous and movement over
1 The interpretation of the passage 240 a 4-18 is elaborately discussed
by R. K. Gaye in the Journal of Philology, xxxi, 1910, pp. 95-116. It is
a question whether in the above quotation Aristotle means that Zeno
argued that half the given time would be equal to double the half, i. e.
the whole time simply, or to double the whole, i. e. four times the half.
Gaye contends (unconvincingly, I think) for the latter.
2 Brochard,- loc. cit., pp. 4, 5.
278 ZENO OF ELEA
lengths which are indeterminate. In the first and third there
is only one moving object, and it is shown that it cannot even
begin to move. The second and fourth, comparing the motions
of two objects, make the absurdity of the hypothesis even
more palpable, so to speak, for they prove that the movement,
even if it has once begun, cannot continue, and that relative
motion is no less impossible than absolute motion. The first
two establish the impossibility of movement by the nature of
space, supposed continuous, without any implication that time
is otherwise than continuous in the same way as space ; in the
last two it is the nature of time (considered as made up of
indivisible elements or instants) which serves to prove the
impossibility of movement, and without any implication that
space is not likewise made up of indivisible elements or points.
The second argument is only another form of the first, and
the fourth rests on the same principle as the third. Lastly, the
first pair proceed on the hypothesis that continuous magni-
tudes are divisible ad infinitum; the second pair give the
other horn of the dilemma, being directed against the assump-
tion that continuous magnitudes are made up of indivisible
elements, an assumption which would scarcely suggest itself
to the imagination until the difficulties connected with the
other were fully realized. Thus the logical order of the argu-
ments corresponds exactly to the historical order in which
Aristotle has handed them down and which was certainly the
order adopted by Zeno.
Whether or not the paradoxes had for Zeno the profound
meaning now claimed for them, it is clear that they have
been very generally misunderstood, with the result that the
criticisms directed against them have been wide of the mark.
Aristotle, it is true, saw that the first two arguments, the
Dichotomy and the Achilles, come to the same thing, the latter
differing from the former only in the fact that the ratio of
each space traversed by Achilles to the preceding space is not
that of 1 : 2 but a ratio of 1 : n, where n may be any number,
however large ; but, he says, both proofs rest on the fact that
a certain moving object ' cannot reach the end of the course if
the magnitude is divided in a certain way'. 1 But another
passage shows that he mistook the character of the argument
1 Arist. Phys. vi. 9, 239 b 18-24.
ZENO'S ARGUMENTS ABOUT MOTION 279
in the Dichotomy. He observes that time is divisible in
exactly the same way as a length ; if therefore a length is
infinitely divisible, so is the corresponding time; he adds
' this is why (Sio) Zeno's argument falsely assumes that it is
not possible to traverse or touch each of an infinite number of
points in a finite time \ l thereby implying that Zeno did not
regard time as divisible ad infinitum like space. Similarly,
when Leibniz declares that a space divisible ad infinitum
is traversed in a time divisible ad infinitum, he, like Aristotle,
is entirely beside the question. Zeno was perfectly aware that,
in respect of divisibility, time and space have the same
property, and that they are alike, always, and concomitantly,
divisible ad infinitum. The question is how, in the one as
in the other, this series of divisions, by definition inexhaustible,
can be exhausted ; and it must be exhausted if motion is to
be possible. It is not an answer to say that the two series
are exhausted simultaneously.
The usual mode of refutation given by mathematicians
from Descartes to Tannery, correct in a sense, has an analogous
defect. To show that the sum of the infinite series 1 + -| + J + . . .
is equal to 2, or to calculate (in the Achilles) the exact moment
when Achilles will overtake the tortoise, is to answer the
question whe >i? whereas the question actually asked is how%
On the hypothesis of divisibility ad infinitum you will, in the
Dichotomy, never reach the limit, and, in the Achilles, the
distance separating Achilles from the tortoise, though it con-
tinually decreases, will never vanish. And if you introduce
the limit, or, with a numerical calculation, the discontinuous,
Zeno is quite aware that his arguments are no longer valid.
We are then in presence of another hypothesis as to the com-
position of the continuum ; and this hypothesis is dealt with
in the third and fourth arguments. 2
It appears then that the first and second arguments, in their
full significance, were not really met before G. Cantor formu-
lated his new theory of continuity and infinity. On this I
can only refer to Chapters xlii and xliii of Mr. Bertrand
Russell's Principles of Mathematics, vol. i. Zeno's argument
in the Dichotomy is that, whatever motion we assume to have
taken place, this presupposes another motion ; this in turn
1 lb. vi. 2, 233 a 16-23. 2 Brochard, loc. cit., p. 9.
280 ZENO OF ELEA
another, and so on ad infinitum. Hence there is an endless
regress in the mere idea of any assigned motion. Zeno's
argument has then to be met by proving that the 'infinite
regress ' in this case is ' harmless '.
As regards the Achilles, Mr. G. H. Hardy remarks that ' the
kernel of it lies in the perfectly valid proof which it affords
that the tortoise passes through as many points as Achilles,
a view which embodies an accepted doctrine of modern mathe-
matics V
The argument in the Arrow is based on the assumption that
time is made up of indivisible elements or instants. Aristotle
meets it by denying the assumption. ' For time is not made
up of indivisible instants (nows), any more than any other
magnitude is made up of indivisible elements.' ' (Zeno's result)
follows through assuming that time is made up of (indivisible)
instants (noivs) ; if this is not admitted, his conclusion does
not follow.' 2 On the other hand, the modern view is that
Zeno's contention is true : ' If ' (said Zeno) ' everything is at
rest or in motion when it occupies a space equal to itself, and
if what moves is always in the instant, it follows that the
moving arrow is unmoved.' Mr. Russell 3 holds that this is
' a very plain statement of an elementary fact ' ;
' it is a very important and very widely applicable platitude,
namely " Every possible value of a variable is a constant ".
If # be a variable which can take all values from to 1,
all the values it can take are definite numbers such as -| or \ ,
which are all absolute constants . . . Though a variable is
always connected with some class, it is not the class, nor
a particular member of the class, nor yet the whole class, but
any member of the class.' The usual x in algebra * denotes
the disjunction formed by the various members' . . . 'The
values of x are then the terms of the disjunction ; and each
of these is a constant. This simple logical fact seems to
constitute the essence of Zeno's contention that the arrow
is always at rest.' ' But Zeno's argument contains an element
which is specially applicable to continua. In the case of
motion it denies that there is such a thing as a state of motion.
In the general case of a continuous variable, it may be taken
as denying actual infinitesimals. For infinitesimals are an
1 Encyclopaedia Britannica, art. Zeno.
2 Arist. Phys. vi. 9, 239 b 8, 31.
3 Russell, Principles of Mathematics, i, pp. 350, 351.
ZENO'S ARGUMENTS ABOUT MOTION 281
attempt to extend to the values of a variable the variability
which belongs to it alone. When once it is firmly realized
that all the values of a variable are constants, it becomes easy
to see, by taking any two such values," that their difference is
always finite, and hence that there are no infinitesimal differ-
ences. If x be a variable which may take all real values
from to 1, then, taking any two of these values, we see that
their difference is finite, although x is a continuous variable.
It is true the difference might have been less than the one we
chose ; but if it had been, it would still have been finite. The
lower, limit to possible differences is zero, but all possible
differences are finite ; and in this there is no shadow of
contradiction. This static theory of the variable is due to the
mathematicians, and its absence in Zeno's day led him to
suppose that continuous change was impossible without a state
of change, which involves infinitesimals and the contradiction
of a body's being where it is not.'
In his later chapter on Motion Mr. Russell concludes as
follows : 1
' It is to be observed that, in consequence of the denial
of the infinitesimal and in consequence of the allied purely
technical view of the derivative of a function, we must
entirely reject the notion of a state of motion. Motion consists
merely in the occupation of different places at different times,
subject to continuity as explained in Part V. There is no
transition from place to place, no consecutive moment or
consecutive position, no such thing as velocity except in the
sense of a real number which is the limit of a certain set
of quotients. The rejection of velocity and acceleration as
physical facts (i. e. as properties belonging at each instant to
a moving point, and not merely real numbers expressing limits
of certain ratios) involves, as we shall see, some difficulties
in the statement of the laws of motion; but the reform
introduced by Weierstrass in the infinitesimal calculus has
rendered this rejection imperative.'
We come lastly to the fourth argument (the Stadium).
Aristotle's representation of it is obscure through its extreme
brevity of expression, and the matter is further perplexed by
an uncertainty of reading. But the meaning intended to be
conveyed is fairly clear. The eight A% B's and O's being
1 Op. cit, p. 473.
282
ZENO OF ELEA
initially in the position shown in Figure 1, suppose, e.g., that
the B's move to the right and the O's to the left with equal
A,
A 2
A 3
A 4
h.
A e
A r
A,
B 8
B,
B 6
B 5
*1
B 9
B ?
B,
■ N-
r-
c,
C ?
C .l
c 4
C5l C 6
cj
C 8
velocity until the rows are vertically under one another as in
Figure 2. Then C x has passed alongside all the eight B's (and B x
Ai
A 2
A 3
A 4
A 5
A*
A r
A 8
B 8
B r
B 6
Bs
B 4
Bj
B 2
B,
Ci
c 2
C 3
C t
c 5
c 6
C T
c 8
alongside all the eight O's), while B x has passed alongside only
half the A' a (and similarly for G x ). But (Aristotle makes Zeno
say) C x is the same time in passing each of the B's as it is in
'passing each of the A's. It follows that the time occupied by C x
in passing all the A's is the same as the time occupied by
C\ in passing half the A's, or a given time is equal to its half.
Aristotle's criticism on this is practically that Zeno did not
"understand the difference between absolute and relative motion.
This is, however, incredible, and another explanation must be
found. The real explanation seems to be that given by
A 1 A a |A a |A 4 |A s lA 6 |A^A fl
B 8
B,
B 6
Bs
B 4
B *
B g
B 1
C.j C 2 C 3 C^ C 5 Cq Cy Cg
Brochard, Noel and Russell. Zeno's object is to prove that
time is not made up of indivisible elements or instants.
Suppose the B's have moved one place to the right and the O's
one place to the left, so that B x , which was under A v is now
under A 5 , and C x , which was under A 5 , is now under J. 4 . We
must suppose that B x snid C\ are absolute indivisible elements
of space, and that they move to their new positions in an
ZENO'S ARGUMENTS ABOUT MOTION 283
instant, the absolute indivisible element of time ; this is Zeno's
hypothesis. But, in order that B 1} C x may have taken up
their new positions, there must have been a moment at which
they crossed or B l was vertically over C x . Yet the motion
has, by hypothesis, taken place in an indivisible instant.
Therefore, either they have not crossed (in which case there
is no movement), or in the particular indivisible instant two
positions have been occupied by the two moving objects, that
is to say, the instant is no longer indivisible. And, if the
instant is divided into two equal parts, this, on the hypothesis
of indivisibles, is equivalent to saying that an instant is double
of itself.
Two remarks may be added. Though the first two argu-
ments are directed against those who assert the divisibility ad
infinitum of magnitudes and times, there is no sufficient
justification for Tannery's contention that they were specially
directed against a view, assumed by him to be Pythagorean,
that bodies, surfaces and lines are made up of mathematical
points. There is indeed no evidence that the Pythagoreans
held this view at all ; it does not follow from their definition
of a point as a ' unit having position ' (fiovas Oio-iv tyovcra) ;
and, as we have seen, Aristotle says that the Pythagoreans
maintained that units and numbers have magnitude. 1
It would appear that, after more than 2,300 years, con-
troversy on Zeno's arguments is yet by no means at an end.
But the subject cannot here be pursued further. 2
1 Arist. Metaph. M. 6, 1080 b 19, 32.
2 It is a pleasure to be able to refer the reader to a most valuable and
comprehensive series of papers by Professor Florian Cajori, under the
title 'The History of Zeno's arguments on Motion', published in the
American Mathematical Monthly of 1915, and also available in a reprint.
This work carries the history of the various views and criticisms of
Zeno's arguments down to 1914. I may also refer to the portions of
Bertrand Russell's work, Our Knowledge of the External World as a Field
for Scientific Method in Philosophy, 1914, which deal with Zeno, and to
Philip E. B. Jourdain's article, ' The Flying Arrow ; an Anachronism ', in
Mind, January 1916, pp. 42-55.
IX
PLATO
It is in the Seventh Book of the Republic that we find
the most general statement of the attitude of Plato towards
mathematics. Plato regarded mathematics in its four branches,
arithmetic, geometry, stereometry and astronomy, as the first
essential in the training of philosophers and of those who
should rule his ideal State ; ' let no one destitute of geometry
enter my doors', said the inscription over the door of his
school. There could be no better evidence of the supreme
importance which he attached to the mathematical sciences.
What Plato emphasizes throughout when speaking of mathe-
matics is its value for the training of the mind ; its practical
utility is of no account in comparison. Thus arithmetic must
be pursued for the sake of knowledge, not for any practical
ends such as its use in trade * ; the real science of arithmetic
has nothing to do with actions, its object is knowledge. 2
A very little geometry and arithmetical calculation sufiices
for the commander of an army ; it is the higher and more
advanced portions which tend to lift the mind on high and
to enable it ultimately to see the final aim of philosophy,
the idea of the Good 3 ; the value of the two sciences consists
in the fact that they draw the soul towards truth and create
the philosophic attitude of mind, lifting on high the things
which our ordinary habit would keep down. 4
The extent to which Plato insisted on the purely theoretical
character of the mathematical sciences is illustrated by his
peculiar views about the two subjects which the ordinary
person would regard as having, at least, an important practical
side, namely astronomy and music. According to Plato, true
astronomy is not concerned with the movements of the visible
1 Rep. vii. 525 c, d. 2 Politicus 258 d.
3 Rep. 526 d, e. 4 lb. 527 b.
PLATO 285
heavenly bodies. The arrangement of the stars in the heaven
and their apparent movements are indeed wonderful and
beautiful, but the observation of and the accounting for them
falls far short of true astronomy. Before we can attain to
this we must get beyond mere observational astronomy, 'we
must leave the heavens alone '. The true science of astronomy
is in fact a kind of ideal kinematics, dealing with the laws
of motion of true stars in a sort of mathematical heaven of
which the visible heaven is an imperfect expression in time
and space. The visible heavenly bodies and their apparent
motions we are to regard merely as illustrations, comparable
to the diagrams which the geometer draws to illustrate the
true straight lines, circles, &c, about which his science reasons ;
they are to be used as ' problems ' only, with the object of
ultimately getting rid of the apparent irregularities and
arriving at 'the true motions with which essential speed
and essential slowness move in relation to one another in the
true numbers and the true forms, and carry their contents
with them ' (to use Burnet's translation of ra ei/ovra). 1
1 Numbers ' in this passage correspond to the periods of the
apparent motions; the 'true forms' are the true orbits con-
trasted with the apparent. It is right to add that according
to one view (that of Burnet) Plato means, not that true
astronomy deals with an ' ideal heaven ' different from the
apparent, but that it deals with the true motions of the visible
bodies as distinct from their apparent motions. This would
no doubt agree with Plato's attitude in the Laws, and at the
time when he set to his pupils as a problem for solution
the question by what combinations of uniform circular revolu-
tions the apparent movements of the heavenly bodies can be
accounted for. But, except on the assumption that an ideal
heaven is meant, it is difficult to see what Plato can mean
by the contrast which he draws between the visible broideries
of heaven (the visible stars and their arrangement), which
are indeed beautiful, and the true broideries which they
only imitate and which are infinitely more beautiful and
marvellous.
This was not a view of astronomy that would appeal to
the ordinary person. Plato himself admits the difficulty.
1 Rep. vii. 529 c-530 c.
286 PLATO
When Socrates's interlocutor speaks of the use of astronomy
for distinguishing months and seasons, for agriculture and
navigation, and even for military purposes, Socrates rallies
him on his anxiety that his curriculum should not consist
of subjects which the mass of people would regard as useless :
' it is by no means an easy thing, nay it is difficult, to believe
that in studying these subjects a certain organ in the mind
of every one is purified and rekindled which is destroyed and
blinded by other pursuits, an organ which is more worthy
of preservation than ten thousand eyes ; for by it alone is
truth discerned.' *
As with astronomy, so with harmonics. 2 The true science of
harmonics differs from that science as commonly understood.
Even the Pythagoreans, who discovered the correspondence
of certain intervals to certain numerical ratios, still made
their theory take too much account of audible sounds. The
true science of harmonics should be altogether independent
of observation and experiment. Plato agreed with the Pytha-
goreans as to the nature of sound. Sound is due to concussion of
air, and when there is rapid motion in the air the tone is high-
pitched, when the motion is slow the tone is low; when the
speeds are in certain arithmetical proportions, consonances or
harmonies result. But audible movements produced, say, by
different lengths of strings are only useful as illustrations;
they are imperfect representations of those mathematical
movements which produce mathematical consonances, and
it is these true consonances which the true apfioviKos should
study.
We get on to easier ground when Plato discusses geometry.
The importance of geometry lies, not in its practical use, but
in the fact that it is a study of objects eternal and unchange-
able, and tends to lift the soul towards truth. The essence
of geometry is therefore directly opposed even to the language
which, for want of better terms, geometers are obliged to use ;
thus they speak of ' squaring ', ' applying (a rectangle) ',
'.adding ', &c, as if the object were to do something, whereas
the true purpose of geometry is knowledge. 3 Geometry is
concerned, not with material things, but with mathematical
1 Rep. 527 d, e. 2 lb. 531 A-c.
3 lb. vii. 526 D-527 b.
PLATO 287
points, lines, triangles, squares, &c, as objects of pure thought.
If we use a diagram in geometry, it is only as an illustration ;
the triangle which we draw is an imperfect representation
of the real triangle of which we think. Constructions, then,
or the processes of squaring, adding, and so on, are not of the
essence of geometry, but are actually antagonistic to it. With
these views before us, we can without hesitation accept as
well founded the story of Plutarch that Plato blamed Eudoxus,
Archytas and Menaechmus for trying to reduce the dupli-
cation of the cube to mechanical constructions by means of
instruments, on the ground that 'the good of geometry is
thereby lost and destroyed, as it is brought back to things
of sense instead of being directed upward and grasping at
eternal and incorporeal images '. 1 It follows almost inevitably
that we must reject the tradition attributing to Plato himself
the elegant mechanical solution of the problem of the two
mean proportionals which we have given in the chapter on
Special Problems (pp. 256-7). Indeed, as we said, it is certain
on other grounds that the so-called Platonic solution was later
than that of Eratosthenes ; otherwise Eratosthenes would
hardly have failed to mention it in his epigram, along
with the solutions by Archytas and Menaechmus. Tannery,
indeed, regards Plutarch's story as an invention based on
nothing more than the general character of Plato's philosophy,
since it took no account of the real nature of the solutions
of Archytas and Menaechmus; these solutions are in fact
purely theoretical and would have been difficult or impossible
to carry out in practice, and there is no reason to doubt that
the solution by Eudoxus was of a similar kind. 2 This is true,
but it is evident that it was the practical difficulty quite as
much as the theoretical elegance of the constructions which
impressed the Greeks. Thus the author of the letter, wrongly
attributed to Eratosthenes, which gives the history of the
problem, says that the earlier solvers had all solved the
problem in a theoretical manner but had not been able to
reduce their solutions to practice, except to a certain small
extent Menaechmus, and that with difficulty ; and the epigram
of Eratosthenes himself says, ' do not attempt the impracticable
1 Plutarch, Qitaest. Conviv. viii. 2. 1, p. 718 F.
2 Tannery, La geometrie grecque, pp. 79, 80.
288 PLATO
business of the cylinders of Archytas or the cutting of the
cone in the three curves of Menaechmus '. It would therefore
be quite possible for Plato to regard Archytas and Menaechmus
as having given constructions that were ultra-mechanical, since
they were more mechanical than the ordinary constructions by
means of the straight line and circle ; and even the latter, which
alone are required for the processes of ' squaring ', ' applying
(a rectangle) ' and ' adding ', are according to Plato no part of
theoretic geometry. This banning even of simple constructions
from true geometry seems, incidentally, to make it impossible
to accept the conjecture of Hankel that we owe to Plato the
limitation, so important in its effect on the later development
of geometry, of the instruments allowable in constructions to
the ruler and compasses. 1 Indeed, there are signs that the
limitation began before Plato's time (e.g. this may be the
explanation of the two constructions attributed to Oenopides),
although no doubt Plato's influence would help to keep the
restriction in force ; for other instruments, and the use of
curves of higher order than circles in constructions, were
expressly barred in any case where the ruler and compasses
could be made to serve (cf. Pappus's animadversion on a solu-
tion of a ' plane ' problem by means of conies in Apollonius's
Conies, Book V).
Contributions to the philosophy of mathematics.
We find in Plato's dialogues what appears to be the first
serious attempt at a philosophy of mathematics. Aristotle
says that between sensible objects and the ideas Plato placed
1 things mathematical ' (ret fjcaOrj/iaTiKd), which differed from
sensibles in being eternal and unmoved, but differed again
from the ideas in that there can be many mathematical
objects of the same kind, while the idea is one only ; e. g. the
idea of triangle is one, but there may be any number of
mathematical triangles as of visible triangles, namely the
perfect triangles of which the visible triangles are imper-
fect copies. A passage in one of the Letters (No. 7, to the
friends of Dion) is interesting in this connexion. 2 Speaking
of a circle by way of example, Plato says there is (1) some-
1 Hankel, op. cit., p. 156. 2 Plato, Letters, 342 b, c, 343 A, B.
THE PHILOSOPHY OF MATHEMATICS 289
thing Called a circle and known by that name ; next there
is (2) its definition as that in which the distances from its
extremities in all directions to the centre are always equal,
for this may be said to be the definition of that to which the
names ' round ' and ' circle ' are applied ; again (3) we have
the circle which is drawn or turned : this circle is perishable
and perishes; not so, however, with (4) avros 6 kvkXos, the
essential circle, or the idea of circle : it is by reference to
this that the other circles exist, and it is different from each
of them. The same distinction applies to anything else, e. g.
the straight, colour, the good, the beautiful, or any natural
or artificial object, fire, water, &c. Dealing separately with
the four things above distinguished, Plato observes that there
is nothing essential in (1) the name : it is merely conventional ;
there is nothing to prevent our assigning the name ' straight '
to what we now call ' round ' and vice versa ; nor is there any
real definiteness about (2) the definition, seeing that it too
is made up of parts of speech, nouns and verbs. The circle
(3), the particular circle drawn or turned, is not free from
admixture of other things : it is even full of what is opposite
to the true nature of a circle, for it will anywhere touch
a straight line ', the meaning of which is presumably that we
cannot in practice draw a circle and a tangent with only one
point common (although a mathematical circle and a mathe-
matical straight line touching it meet in one point only). It
will be observed that in the above classification there is no
place given to the many particular mathematical circles which
correspond to those which we draw, and are intermediate
between these imperfect circles and the idea of circle which
is one only.
(a) The hypotheses of mathematics.
The hypotheses of mathematics are discussed by Plato in
the Republic.
' I think you know that those who occupy themselves with
geometries and calculations and the like take for granted the
odd and the even, figures, three kinds of angles, and other
things cognate to these in each subject ; assuming these things
as known, they take them as hypotheses and thenceforward
they do not feel called upon to give any explanation with
1623 XJ
290 PLATO
regard to them either to themselves or any one else, but treat
them as manifest to every one ; basing themselves on these
hypotheses, they proceed at once to go through the rest of
the argument till they arrive, with general assent, at the
particular conclusion to which their inquiry was directed.
Further you know that they make use of visible figures and
argue about them, but in doing so they are not thinking of
these figures but of the things which they represent; thus
it is the absolute square and the absolute diameter which is
the object of their argument, not the diameter which they
draw ; and similarly, in other cases, the things which they
actually model or draw, and which may also have their images
in shadows or in water, are themselves in turn used as
images, the object of the inquirer being to see their abso-
lute counterparts which cannot be seen otherwise than by
thought.' 1
(ft) The two intellectual methods.
Plato distinguishes two processes : both begin from hypo-
theses. The one method cannot get above these hypotheses,
but, treating them as if they were first principles, builds upon
them and, with the aid of diagrams or images, arrives at
conclusions : this is the method of geometry and mathematics
in general. The other method treats the hypotheses as being
really hypotheses and nothing more, but uses them as stepping-
stones for mounting higher and higher until the principle
of all things is reached, a principle about which there is
nothing hypothetical ; when this is reached, it is possible to
descend again, by steps each connected with the preceding
step, to the conclusion, a process which has no need of any
sensible images but deals in ideals only and ends in them 2 ;
this method, which rises above and puts an end to hypotheses,
and reaches the first principle in this way,* is the dialectical
method. For want of this, geometry and the other sciences
which in some sort lay hold of truth are comparable to one
dreaming about truth, nor can they have a waking sight of
it so long as they treat their hypotheses as immovable
truths, and are unable to give any account or explanation
of them. 3
1 Republic, vi. 510 C-E. 2 lb. vi. 510 B 511 A-c.
3 lb. vii. 533 b-e.
THE TWO INTELLECTUAL METHODS 291
With the above quotations we should read a passage of
Proclus.
' Nevertheless certain methods have been handed down. The
finest is the method which by means of analysis carries
the thing sought up to an acknowledged principle ; a method
which Plato, as they say, communicated to Leodamas, and by
which the latter too is said to have discovered many things
in geometry. The second is the method of division, which
divides into its parts the genus proposed for consideration,
and gives a starting-point for the demonstration by means of
the elimination of the other elements in the construction
of what is proposed, which method also Plato extolled as
being of assistance to all sciences.' 1
The first part of this passage, with a like dictum in Diogenes
Laertius that Plato ' explained to Leodamas of Thasos the
method of inquiry by analysis', 2 has commonly been under-
stood as attributing to Plato the invention of the method
of mathematical analysis. But, analysis being according to
the ancient view nothing more than a series of successive
reductions of a theorem or problem till it is finalty reduced
to a theorem or problem already known, it is difficult to
see in what Plato's supposed discovery could have consisted ;
for analysis in this sense must have been frequently used
in earlier investigations. Not only did Hippocrates of Chios
reduce the problem of duplicating the cube to that of finding
two mean proportionals, but it is clear that the method of
analysis in the sense of reduction must have been in use by
the Pythagoreans. On the other hand, Proclus's language
suggests that what he had in mind was the philosophical
method described in the passage of the Republic, which of
course does not refer to mathematical analysis at all ; it may
therefore well be that the idea that Plato discovered the
method of analysis is due to a misapprehension. But analysis
and synthesis following each other are related in the same
way as the upward and downward progressions in the dialec-
tician's intellectual method. It has been suggested, therefore,
that Plato's achievement was to observe the importance
from the point of view of logical rigour, of the confirma-
tory synthesis following analysis. The method of division
1 Proclus, Comm. on Eucl. I, pp. 211. 18-212. 1.
2 Diog. L. iii. 24, p. 74, Cobet.
U 2
292 PLATO
mentioned by Proclus is' the method of successive bipartitions
of genera into species such as we find in the Sophist and
the Pollticus, and has little to say to geometry ; but the
mention of it side by side with analysis itself suggests that
Proclus confused the latter with the philosophical method
referred to.
(y) Definitions.
Among the fundamentals of mathematics Plato paid a good
deal of attention to definitions. In some cases his definitions
connect themselves with Pythagorean tradition ; in others he
seems to have struck out a new line for himself. The division
of numbers into odd and even is one of the most common of
his illustrations ; number, he says, is divided equally, i. e.
there are as many odd numbers as even, and this is the true
division of number ; to divide number (e. g.) into myriads and
what are not myriads is not a proper division. 1 An even
number is defined as a number divisible into two equal parts 2 ;
in another place it is explained as that which is not scalene
but isosceles 3 : a curious and apparently unique application
of these terms to number, and in any case a defective state-
ment unless the term ' scalene ' is restricted to the case in which
one part of the number is odd and the other even ; for of
course an even number can be divided into two unequal odd
numbers or two unequal even numbers (except 2 in the first
case and 2 and 4 in the second). The further distinction
between even-times-even, odd-times-even, even-times-odd and
odd-times-odd occurs in Plato 4 : but, as thrice two is called
odd-times-even and twice three is even-times-odd, the number
in both cases being the same, it is clear that, like Euclid,
Plato regarded even-times-odd and odd-times-even as con-
vertible terms, and did not restrict their meaning in the way
that Nicomachus and the neo-Pythagoreans did.
Coming to geometry we find an interesting view of the
term ' figure '. What is it, asks Socrates, that is true of the
round, the straight, and the other things that you call figures,
and is the same for all ? As a suggestion for a definition
of ' figure ', Socrates says, ' let us regard as figure that which
alone of existing things is associated with colour \ Meno
1 Politicus, 262 D, e. 2 Laws, 895 e.
8 Euthyphro, 12 D. 4 Parmenides, 143 E- 144 A.
DEFINITIONS 293
asks what is to be done if the interlocutor says he docs not
know what colour is; what alternative definition is there?
Socrates replies that it will be admitted that in geometry
there are such things as what we call a surface or a solid,
and so on ; from these examples we may learn what we mean
by figure ; figure is that in which a solid ends, or figure is
the limit (or extremity, irepas) of a solid. 1 Apart from
' figure ' as form or shape, e. g. the round or straight, this
passage makes ' figure ' practically equivalent to surface, and
we are reminded of the Pythagorean term for surface, \poid,
colour or skin, which Aristotle similarly explains as \pcona,
colour, something inseparable from Trepa?, extremity. 2 In
Euclid of course opo$, limit or boundary, is defined as the
extremity (nepas) of a thing, while ' figure ' is that which is
contained by one or more boundaries.
There is reason to believe, though we are not specifically
told, that the definition of a line as ' breadthless length '
originated in the Platonic School, and Plato himself gives
a definition of a straight line as ' that of which the middle
covers the ends ' 3 (i. e. to an eye placed at either end and
looking along the straight line) ; this seems to me to be the
origin of the Euclidean definition ' a line which lies evenly
with the points on it ', which, I think, can only be an attempt
to express the sense of Plato's definition in terms to which
a geometer could not take exception as travelling outside the
subject matter of geometry, i. e. in terms excluding any appeal
to vision. A point had been defined by the Pythagoreans as
a ' monad having position ' ; Plato apparently objected to this
definition and substituted no other ; for, according to Aristotle,
he regarded the genus of points as being a 'geometrical
fiction ', calling a point the beginning of a line, and often using
the term ' indivisible lines ' in the same sense. 4 Aristotle
points out that even indivisible lines must have extremities,
and therefore they do not help, while the definition of a point
as ' the extremity of a line ' is unscientific. 5
The ' round ' (arpoyyvXov) or the circle is of course defined
as ' that in which the furthest points (rot ea-^ara) in all
1 Meno, 75 a-76 a. 2 Arist. De sensu, 439 a 31, &c.
3 Parmenides, 137 E. 4 Arist. Metaph. A. 9, 992 a 20.
5 Arist. Topics, vi. 4, 141 b 21.
294 PLATO
directions are at the same distance from the middle (centre) '}
The ' sphere ' is similarly defined as * that which has the
distances from its centre to its terminations or ends in every
direction equal ', or simply as that which is ' equal (equidistant)
from the centre in all directions \ 2
The Parmenides contains certain phrases corresponding to
what we find in Euclid's preliminary matter. Thus Plato
speaks of something which is ' a part ' but not ' parts ' of the
One, :i reminding us of- Euclid's distinction between a fraction
which is ' a part ', i. e. an aliquot part or submultiple, and one
which is ' parts ', i. e. some number more than one of such
parts, e. g. f. If equals be added to unequals, the sums differ
by the same amount as the original unequals did : 4 an axiom
in a rather more complete form than that subsequently inter-
polated in Euclid.
Summary of the mathematics in Plato.
The actual arithmetical and geometrical propositions referred
to or presupposed in Plato's writings are not such as to suggest
that he was in advance of his time in mathematics ; his
knowledge does not appear to have been more than up to
date. In the following paragraphs I have attempted to give
a summary, as complete as possible, of the mathematics con-
tained in the dialogues.
A proposition in proportion is quoted in the P ar me aides f
namely that, if a > b, then (a + c) : (b + c) < a : b.
In the Laivs a certain number, 5,040, is selected as a most
convenient number of citizens to form a state ; its advantages
are that it is the product of 12, 21 and 20, that a twelfth
part of it is again divisible by 12, and that it has as many as
59 different divisors in all, including all the natural numbers
from 1 to 12 with the exception of 11, while it is nearly
divisible by 11 (5038 being a multiple of 11). G
(a) Regtdar and semi-regular solids.
The 'so-called Platonic figures', by which are meant the
five regular solids, are of course not Plato's discovery, for they
had been partly investigated by the Pythagoreans, and very
1 Parmenides, 137 E. 2 Timaeus, 33 B, 34 B.
3 Parmenides, 153 D. 4 lb. 154 B.
5 lb. 154 d. 6 Laws, 537 E-538 A.
REGULAR AND SEMI-REGULAR SOLIDS 295
fully by Theaetetus ; they were evidently only called Platonic
because of the use made of them in the Timaeus, where the
particles of the four elements are given the shapes of the first
four of the solids, the pyramid or tetrahedron being appro-
priated to fire, the octahedron to air, the icosahedron to water,
and the cube to earth, while the Creator used the fifth solid,
the dodecahedron, for the universe itself. 1
According to Heron, however, Archimedes, who discovered
thirteen semi-regular solids inscribable in a sphere, said that
' Plato also, knew one of them, the figure with fourteen faces,
of which there are two sorts, one made up of eight triangles
and six squares, of earth and air, and already known to some
of the ancients, the other again made up of eight squares and
six triangles, which seems to be more difficult.' 2
The first of these is easily obtained ;' if we take each square
face of a cube and make in it a smaller square by joining
the middle points of each pair of consecutive sides, we get six
squares (one in each face) ; taking the three out of the twenty-
four sides of these squares which are about any one angular
point of the cube, we have an equilateral triangle ; there are
eight of these equilateral triangles, and if we cut off from the
corners of the cube the pyramids on these triangles as bases,
we have a semi-regular polyhedron
inscribable in a sphere and having
as faces eight equilateral triangles
and six squares. The description of
the second semi-regular figure with
fourteen faces is wrong: there are
only two more such figures, (1) the
figure obtained by cutting off from
the corners of the cube smaller
pyramids on equilateral triangular bases such that regular
octagons, and not squares, are left in the six square faces,
the figure, that is, contained by eight triangles and six
octagons, and (2) the figure obtained by cutting off from the
corners of an octahedron equal pyramids with square bases
such as to leave eight regular hexagons in the eight faces,
that is, the figure* contained by six squares and eight hexagons.
1 Timaeus, 55 d-56 b, 55 c.
2 Heron, Definitions, 104, p. 66, Heib.
296
PLATO
(/?) The construction of the regular solids.
Plato, of course, constructs the regular solids by simply
putting together the plane faces. These faces are, he observes,
made up of triangles ; and all triangles are decomposable into
two right-angled triangles. Right-angled triangles are either
(1) isosceles or (2) not isosceles, having the two acute angles
unequal. Of the latter class, which is unlimited in number,
one triangle is the most beautiful, that in which the square on
the perpendicular is triple of the square on the base (i. e. the
triangle^ which is the half of an equilateral triangle obtained
by drawing a perpendicular from a vertex on the opposite
side). (Plato is here Pythagorizing. 1 ) One of the regular
solids, the cube, has its faces (squares) made up of the first
kind of right-angled triangle, the isosceles, four of
them being put together to form the square ; three
others with equilateral triangles for faces, the tetra-
hedron, octahedron and icosahedron, depend upon
the other species of right-angled triangle only,
each face being made up of six (not two) of those right-angled
triangles, as shown in the figure ; the fifth solid, the dodeca-
hedron, with twelve regular pentagons for
faces, is merely alluded to, not described, in
the passage before us, and Plato is aware that
its faces cannot be constructed out of the two
elementary right-angled triangles on which the
four other solids depend. That an attempt was made to divide
the pentagon into a number of triangular elements is clear
from three passages, two in Plutarch 2
and one in Alcinous. 3 Plutarch says
that each of the twelve faces of a
dodecahedron is made up of thirty
elementary scalene triangles which are
different from the elementary triangle
of the solids with triangular faces.
Alcinous speaks of the 360 elements
which are produced when each pen-
tagon is divided into five isosceles triangles and each of the
1 Cf. Speusippus in Theol. Ar., p. 61, Ast.
2 Plutarch, Quaest. Plat. 5. 1, 1003 d ; De defectu Oraculorum, c. 33, 428 a.
3 Alcinous, De Doctrina Platonis, c. 11.
THE REGULAR SOLIDS 297
latter into six scalene triangles. If we draw lines in a pen-
tagon as shown in the accompanying figure, we obtain such
a set of triangles in a way which also shows the Pythagorean
pentagram (cf. p. 161, above).
(y) Geometric means between two square numbers
or two cubes.
In the Timaeus Plato, speaking of numbers ' whether solid
or square ' with a (geometric) mean or means between them,
observes that between planes one mean suffices, but to connect
two solids two means are necessary. 1 By planes and solids
Plato probably meant square and cube numbers respectively,
so that the theorems quoted are probably those of Eucl. VIII.
11, 12, to the effect that between two square numbers there is
one mean proportional number, and between two cube numbers
two mean proportional numbers. Nicomachus quotes these
very propositions as constituting ' a certain Platonic theorem \ 2
Here, too, it may be that the theorem is called ' Platonic ' for
the sole reason that it is quoted by Plato in the Timaeus ;
it may well be older, for the idea of two mean proportionals
between two straight lines had already appeared in Hippo-
crates's reduction of the problem of doubling the cube. Plato's
allusion does not appear to be to the duplication of the cube
in this passage any more than in the expression Kvficov av^rj,
' cubic increase ', in the Republic? which appears to be nothing
but the addition of the third dimension to a square, making
a cube (cf . Tptrrj av£r], ' third increase ', 4 meaning a cube
number as compared with 8vi>a/ii?, a square number, terms
which are applied, e.g. to the numbers 729 and 81 respec-
tively).
(8) The two geometrical passages in the Meno.
We come now to the two geometrical passages in the Meno.
In the first fj Socrates is trying to show that teaching is only
reawaking in the mind of the learner the memory of some-
thing. He illustrates by putting to the slave a carefully
prepared series of questions, each requiring little more than
1 Timaeus, 31 c-32 b. 2 Nicom. ii. 24. 6.
3 Republic, 528 b. 4 lb. 587 d.
5 Meno, 82 b-85 b.
298
PLATO
H
/ C \
— ^ c F /
M
N
K
' yes ' or ' no ' for an answer, but leading up to the geometrical
construction of </2. Starting with a straight line AB 2 feet
long, Socrates describes a square ABGD upon it and easily
shows that the area is 4 square feet. Producing the sides
AB, AD to G, K so that BG, DK are equal to AB, AD, and
completing the figure, we have a square of side 4 feet, and this
square is equal to four times the original square and therefore
has an area of 16 stjuare feet. Now, says Socrates, a square
8 feet in area must have its side
greater than 2 and less than 4 feet.
The slave suggests that it is 3 feet
in length. By taking N the
middle point of DK (so that AN
is 3 feet) and completing the square
on AN, Socrates easily shows that
the square on AN is not 8 but 9
square feet in area. If L, M be
the middle points of GH, UK and
GL, GM be joined, we have four
squares in the figure, one of which is ABGD, while each of the
others is equal to it. If now we draw the diagonals BL, LM,
AID, DB of the four squares, each diagonal bisects its square,
and the four make a square BLMD, the area of which is half
that of the square A GHK, and is therefore 8 square feet ;
BL is a side of this square. Socrates concludes with the
words :
'The Sophists call this straight line (BD) the diameter
(diagonal) ; this being its name, it follows that the square
which is double (of the original square) has to be described on
the diameter.'
The other geometrical passage in the Meno is much more
difficult, 1 and it has gathered round it a literature almost
comparable in extent to the volumes that have been written
to explain the Geometrical Number of the Republic. C. Blass,
writing in 1861, knew thirty different interpretations; and
since then many more have appeared. Of recent years
Benecke's interpretation 2 seems to have enjoyed the most
1 Meno, 86 E-87 c.
2 Dr. Adolph Benecke, Ueber die geometrische Hypothesis in Platon's
Menon (Elbing, 1867). See also below, pp. 302-3.
TWO GEOMETRICAL PASSAGES IN THE ME NO 299
acceptance ; nevertheless, I think that it is not the right one,
but that the essentials of the correct interpretation were given
by S. H. Butcher 1 (who, however, seems to have been com-
pletely anticipated by E. F. August, the editor of Euclid, in
1829). It is necessary to begin with a literal translation of
the passage. Socrates is explaining a procedure 'by way
of hypothesis ', a procedure which, he observes, is illustrated
by the practice of geometers
' when they are asked, for example, as regards a given area,
whether it is possible for this area to be inscribed in the form
of a triangle in a given circle. The answer might be, " I do
not yet know whether this area is such as can be so inscribed,
but I think I can suggest a hypothesis which will be useful for
the purpose ; I mean the following. If the given area is such
as, when one has applied it (as a rectangle) to the given
straight line in the circle [ttju SoBtTo-av avrov ypafifxrju, the
given straight line in if, cannot, I think, mean anything
but the diameter of the circle 2 ], it is deficient by a figure
(rectangle) similar to the very figure which is applied, then
one alternative seems to me to result, while again another
results if it is impossible for what I said to be done with it.
Accordingly, by using a hypothesis, I am ready to tell you what
results with regard to the inscribing of the figure in the circle,
namely, whether the problem is possible or impossible."
Let AEB be a circle on AB as diameter, and let AG be the
tangent at A. Take E any point on the circle and draw
ED perpendicular to AB. Complete the rectangles AGED,
EDBF.
Then it is clear that the rectangle GEDA is ' applied ' to
the diameter AB, and also that it ' falls short ' by a figure, the
rectangle EDBF, similar to the ' applied ' rectangle, for
AD:DE = ED:DB.
Also, if ED be produced to meet the circle again in G,
AEG is an isosceles triangle bisected by the diameter AB,
and therefore equal in area to the rectangle AGED.
If then the latter rectangle, ' applied ' to AB in the manner
1 Journal of Philology, vol. xvii, pp. 219-25 ; cf. E. S. Thompson's edition
of the Meno.
2 The obvious 'line' of a circle is its diameter, just as, in the first
geometrical passage about the squares, the ypayL^ the ' line ', of a square
is its side.
300
PLATO
described, is equal to the given area, that area is inscribed in
the form of a triangle in the given circle. 1
In order, therefore, to inscribe in the circle an isosceles
triangle equal to the given area (X), we have to find a point E
on the circle such that, if ED be drawn perpendicular to A B,
c
E
F
fi
1*
Sv J °
D
D'l
B
the rectangle AD . DE is equal to the given area X (' applying '
to AB ^ rectangle equal to X and falling short by a figure
similar to the ' applied ' figure is only another way of ex-
pressing it). Evidently E lies on the rectangular hyperbola
1 Butcher, after giving the essentials of the interpretation of the
passage quite correctly, finds a difficulty. 'If, he says, 'the condition'
(as interpreted by him) 'holds good, the given ^coptov can be inscribed in
a circle. But the converse proposition is not true. The \o»plov can still
be inscribed, as required, even if the condition laid down is not fulfilled ;
the true and necessary condition being that the given area is not greater
than that of the equilateral triangle, i.e. the maximum triangle, which
can be inscribed in the given circle.' The difficulty arises in this way.
Assuming (quite fairly) that the given area is given in the form of a rect-
angle (for any given rectilineal figure can be transformed into a rectangle
of equal area), Butcher seems to suppose that it is identically the given
rectangle that is applied to AB. But this is not necessary. The termi-
nology of mathematics was not quite fixed in Plato's time, and he allows
himself some latitude of expression, so that we need not be surprised to
find him using the phrase ' to apply the area (x<»pioi>) to a given straight
line ' as short for ' to apply to a given straight line a rectangle equal (but not
similar) to the given area ' (cf. Pappus vi, p. 544. 8-10 p.rj -nav rb boOiv
napa rqv boOeicrav irapafiaWeaOat (XXdnov rerpayooixo, ' that it is not every
given (area) that can be applied (in the form of a rectangle) falling short
by a square figure'). If we interpret the expression in this way, the
converse is true ; if we cannot apply, in the way described, a rectangle
equal to the given rectangle, it is because the given rectangle is greater
than the equilateral, i.e. the maximum, triangle that can be inscribed in
the circle, and the problem is therefore impossible of solution. (It was
not till long after the above was written that my attention was drawn to
the article on the same subject in the Journal of Philology, xxviii, 1903,
pp. 222-40, by Professor Cook Wilson. I am gratified to find that my
interpretation of the passage agrees with his.)
TWO GEOMETRICAL PASSAGES IN THE MEND 301
the equation of which referred to AB, AC as axes of x, y is
xy — b 2 , where b 2 is equal to the given area. For a real
solution it is necessary that b 2 should be not greater than the
equilateral triangle inscribed in the circle, i. e. not greater than
3 Vs .a 2 /4, where a is the radius of the circle. If b 2 is equal
to this area, there is only one solution (the hyperbola in that
case touching the circle) ; if b 2 is less than this area, there are
two solutions corresponding to two points E, E' in which the
hyperbola cuts the circle. If AD — x, we have OD = x — a,
BE = \/(2ax — x 2 ), and the problem is the equivalent of
solving the equation
x V(2ax — x 2 ) — b 2 ,
or x 2 (2 ax — x 2 ) = b 4 .
This is an equation of the fourth degree which can be solved
by means of conies, but not by means of the straight line
and circle. The solution is given by the points of intersec-
tion of the hyperbola xy = b L and the circle y 2 = 2 ax — x 2 or
x 2 + y 2 = 2 ax. In this respect therefore the problem is like
that of finding the two mean proportionals, which was likewise
solved, though not till later, by means of conies (Menaechmus).
I am tempted to believe that we have here an allusion to
another actual problem, requiring more than the straight
line and circle for its solution,
which had exercised the minds
of geometers by the time of
Plato, the problem, namely, of
inscribing in a circle a triangle
equal to a given area, a problem
which was still awaiting a
solution, although it had been
reduced to the problem of
applying a rectangle satisfying the condition described by
Plato, just as the duplication of the cube had been reduced
to the problem of finding two mean proportionals. Our
problem can, like the latter problem, easily be solved by the
* mechanical ' use of a ruler. Suppose that the given rectangle
is placed so that the side AD lies along the diameter A B of
the circle. Let E be the angle of the rectangle AD EC opposite
to A. Place a ruler so that it passes through E and turn
302
PLATO
it about E until it passes through a point P of the circle such
that, if EP meets AB and AG produced in T, R, PT shall be
equal to ER. Then, since RE = PT, AB = MT, where M is
the foot of the ordinate PM.
Therefore DT = AM, and
AM:AB = BT:MT
whence
= EB:P3I,
PAL MA = ED.DA,
and APM is the half of the required (isosceles) triangle.
Benecke criticizes at length the similar interpretation of the
passage given by E. F. August. So far, however, as his objec-
tions relate to the translation of particular words in the
Greek text, they are, in my opinion, not well founded. 1 .For
the rest, Benecke holds that, in view of the difficulty of the
problem which emerges, Plato is unlikely to have introduced
it in such an abrupt and casual way into the conversation
between Socrates and Meno. But the problem is only one
of the same nature as that of the finding of two mean
proportionals which was already a famous problem, and, as
regards the form of the allusion, it is to be noted that Plato
was fond of dark hints in things mathematical.
If the above interpretation is too difficult (which I, for one,
do not admit), Benecke's is certainly too easy. He connects
his interpretation of the passage with the earlier passage
about the square of side 2 feet ; according to him the problem
is, can an isosceles right-angled tri-
angle equal to the said square be
inscribed in the given circle? This
is of course only possible if the
radius of the circle is 2 feet in length.
If AB, BE be two diameters at right
angles, the inscribed triangle is ABE;
the square AG BO formed by the radii
A0, OB and the tangents at B, A
is then the ' applied ' rectangle, and
the rectangle by which it falls short is also a square and equal
1 The main point of Benecke's criticisms under this head has reference
to TOiovroi ^oopia) olov in the phrase iWe'nreiv roiotrrcp ^oopio) olov av avrb to
7rapaT.T(ifj,ivov ft. He will have it that toiovt<o olov cannot mean ' similar to ',
TWO GEOMETRICAL PASSAGES IN THE MENU 303
to the other square. If this were the correct interpretation,
Plato is using much too general language about the applied
rectangle and that by which it is deficient; it would be
extraordinary that he should express the condition in this
elaborate way when he need only have said that the radius
of the circle must be equal to the side of the square and
therefore 2 feet in length. The explanation seems to me
incredible. The criterion sought by Socrates is evidently
intended to be a real Siopio-uos, or determination of the
conditions or limits of the possibility of a solution of the pro-
blem whether in its original form or in the form to which
it is reduced ; but it is no real Siopiarfios to say what is
equivalent to saying that the problem is possible of solution
if the circle is of a particular size, but impossible if the circle
is greater or less than that size.
The passage incidentally shows that the idea of a formal
SiopKr/169 defining the limits of possibility of solution was
familiar even before Plato's time, and therefore that Proclus
must be in error when he says that Leon, the pupil of
Neoclides, ' invented Stopio-fioi (determining) when the problem
which is the subject of investigation is possible and when
impossible ',* although Leon may have been the first to intro-
duce the term or to recognize formally the essential part
played by SiopLcr/jioi in geometry.
(e) Plato and the doubling of the cube.
The story of Plato's relation to the problem of doubling
the cube has already been told (pp. 245-6, 255). Although the
solution attributed to him is not his, it may have been with
this problem in view that he complained that the study of
solid geometry had been unduly neglected up to his time. 2
and he maintains that, if Plato had meant it in this sense, he should
have added that the ' defect ', although ' similar ', is not similarly situated.
I see no force in this argument in view of the want of fixity in mathe-
matical terminology in Plato's time, and of his own habit of varying his
phrases for literary effect. Benecke makes the words mean ' of the same
kind \ e. g. a square with a square or a rectangle with a rectangle. But
this would have no point unless the figures are squares, which begs the
whole question.
1 Proclus on Eucl. I, p. 66. 20-2.
2 Republic, vii. 528 A-c.
304 PLATO
(£) Solution of x 2 + y 2 = z 2 in integers.
We have already seen (p. 81) that Plato is credited with
a rule (complementary to the similar rule attributed to Pytha-
goras) for finding a whole series of square numbers the sum
of which is also a square ; the formula is
(2n) 2 +(n 2 -l) 2 = (n 2 +l) 2 .
(r}) Incommensuvables.
On the subject of incommensurables or irrationals we have
first the passage of the Theaetetus record in that Theodoras
proved the incommensurability of Vz, \/5 ... Vl7, after
which Theaetetus generalized the theory of such ' roots '.
This passage has already been fully discussed (pp. 203-9).
The subject of incommensurables comes up again in the Laws,
where Plato inveighs against the ignorance prevailing among
the Greeks of his time of the fact that lengths, breadths and
depths may be incommensurable as well as commensurable
with one another, and appears to imply that he himself had
not learnt the fact till late (aKotxras oyjri wore), so that he
was ashamed for himself as well as for his countrymen in
general. 1 But the irrationals known to Plato included more
than mere ' surds ' or the sides of non-squares ; in one place
he says that, just as an even number may be the sum of
either two odd or two even numbers, the sum of two irra-
tionals may -be either rational or irrational. 2 An obvious
illustration of the former case is afforded by a rational straight
line divided ' in extreme and mean ratio '. Euclid (XIII. 6)
proves that each of the segments is a particular kind of
irrational straight line called by him in Book X an apotome ;
and to suppose that the irrationality of the two segments was
already known to Plato is natural enough if we are correct in
supposing that ' the theorems which' (in the words of Proclus)
' Plato originated regarding the section ' 3 were theorems about
what came to be called the ' golden section ', namely the
division of a straight line in extreme and mean ratio as in
Eucl. II. 11 and VI. 30. The appearance of the latter problem
in Book II, the content of which is probably all Pythagorean,
suggests that the incommensurability of the segments with
1 Laws, 819 D-820 c. 2 Hippias Maior, 303 b, C.
3 Proclus on Eucl. I, p. 67. 6.
INCOMMENSURABLES 305
the whole line was discovered before Plato's time, if not as
early as the irrationality of V2.
(0) The Geometrical Number.
This is not the place to discuss at length the famous passage
about the Geometrical Number in the Republic. 1 Nor is its
mathematical content of importance ; the whole thing is
mystic rather than mathematical, and is expressed in
rhapsodical language, veiling by fanciful phraseology a few
simple mathematical conceptions. The numbers mentioned
are supposed to be two. Hultsch and Adam arrive at the
same two numbers, though by different routes. The first
of these numbers is 216, which according to Adam is the sum
of three cubes 3 3 + 4 3 + 5 3 ; 2 3 . 3 3 is the form in which
Hultsch obtains it. 2
1 Republic, viii. 546 b-d. The number of interpretations of this passage
is legion. For an exhaustive discussion of the language as well as for
one of the best interpretations that has been put forward, see Dr. Adam's
edition of the Republic, vol. ii, pp. 204-8, 264-312.
2 The Greek is eV <o npairco av^qaat bvvapevai re Kal BvvaaTevo/jievai, rpels
dnocTTacreis, Terrapas 8e opovs Xa/3o{5(rai 6p.oiovvT<ov re Kal avop.oiovvTa>v Kal
av£6vTO)v Kal (pOivovrcov, navra Trpnar^yopa Kal prjra npos aWrjXa aTrttyrjvav,
which Adam translates by ' the first number in which root and
square increases, comprehending three distances and four limits, of
elements that make like and unlike and wax and wane, render all
things conversable and rational with one another \ av^aeis are
clearly multiplications, dwdpeval re ko.1 Swao-Ttvofievai are explained in
this way. A straight line is said dvvaodai ('to be capable of) an area,
e. g. a rectangle, when the square on it is equal to the rectangle ; hence
hwapiivr) should mean a side of a square. dwao-Tevopepij represents a sort
of passive of dvuapeprj, meaning that of which the Swap.ivr) is ' capable ' ;
hence Adam takes it here to be the square of which the 8wapev>] is the
side, and the whole expression to mean the product of a square and its
side, i. e. simply the cube of the side. The cubes 3 3 , 4 3 , 5 3 are supposed
to be meant because the words in the description of the second number
'of which the ratio in its lowest terms 4:3 when joined to 5' clearly
refer to the right-angled triangle 3, 4, 5, and because at least three
authors, Plutarch (Be Is. et Os. 373 F), Proclus (on Eucl. I, p. 428. 1) and
Aristides Quintilianus (De mus., p. 152 Meibom. = p. 90 Jahn) say that
Plato used the Pythagorean or ' cosmic ' triangle in
his Number. The ' three distances ' are regarded a B
as ' dimensions ', and the ' three distances and
four limits ' are held to confirm the interpretation
' cube ', because a solid (parallelepiped) was said to
have 'three dimensions and four limits' (Theol. Ar.,
p. 16 Ast, and Iambi, in Nicom., p. 93. 10), the limits
being bounding points as A, B, C, D in the accom-
panying figure. ' Making like and unlike ' is sup-
posed to refer to the square and oblong forms in which the second
1523 X
306 PLATO
The second number is described thus :
' The ratio 4 : 3 in its lowest terms (* the base ', 7rvd/j.rju, of
the ratio iwiTpiTos) joined or wedded to 5 yields*two harmonies
when thrice increased (rph avgrjOeis), the one equal an equal
number of times, so many times 100, the other of equal length
one way, but oblong, consisting on the one hand of 100 squares
of rational diameters of 5 diminished by one each or, if of
number is stated.
Another view of the whole passage has recently appeared (A. G. Laird,
Plato's Geometrical Number and the comment of Proclus, Madison, Wiscon-
sin, 1918). Like all other solutions, it is open to criticism in some
details, but it is attractive in so far as it makes greater use of Proclus
(in Platonis remp., vol. ii, p. 36 seq. Kroll) and especially of the passage
(p. 40) in which he illustrates the foi'mation of the ' harmonies ' by means
of geometrical figures. According to Mr. Laird there are not tivo separ-
ate numbers, and the description from which Hultsch and Adam derive
the number 216 is not a description of a number but a statement of a
general method of formation of ' harmonies ', which is then applied to
the triangle 3, 4, 5 as a particular case, in order to produce the one
Geometrical Number. The basis of the whole thing is the use of figures
like that of Eucl. VI. 8 (a right-angled triangle divided by a perpendicular
from the right angle on the opposite side into two right-angled triangles
similar to one another and to the original triangle). Let ABC be a
right-angled triangle in which the sides CB, BA containing the right
angle are rational numbers a, b respectively.
Draw AF at right angles to AC meeting CB
produced in F. Then the figure AFC is that of
Eucl. VI. 8, and of course AB 2 =CB.BF.
Complete the rectangle ABFL, and produce
FL, CA to meet at K. Then, by similar tri-
angles, CB, BA, FB (= AL) and KL are four
straight lines in continued proportion, and their
lengths are a, b, b 2 /a, b 3 /a 2 respectively. Mul-
tiplying throughout by a 1 in order to get rid of
fractions, we may take the lengths to be a 3 ,
a 2 b, ab 2 , b 7, respectively. Now, on Mr. Laird's
view, (tirade 8vudfxev(tL are squares, as AB 2 , and
avj-rjaeis Swao-revo/ievai rectangles, as FB, BC, to
which the squares are equal. ' Making like and
unlike ' refers to the equal factors of a 6 , 6 3 and the unequal factors of
a 2 b, ab 2 ; the terms a 3 , a 2 b, ab 2 , b 3 are four terms (opoi) of a continued
proportion with three intervals (anovTaaus), and of course are all ' con-
versable and rational with one another '. (Incidentally, out of such
terms we can even obtain the number 216, for if we put a = 2, b — 3, we
have 8, 12, 18, 27, and the product of the extremes 8 . 27 = the product
of the means 12 . 18 = 216). Applying the method to the triangle 3, 4, 5
(as Proclus does) we have the terms 27, 36, 48, 64, and the first three
numbers, multiplied respectively by 100, give the elements of the
Geometrical Number 3600 2 = 2700 .4800. On this interpretation t P \s
avgqOcis simply means raised to the third dimension or ' made solid ' (as
Aristotle says, Politics (E). 12, 1316 a 8), the factors being of course
3.3.3 = 27, 3.3.4 = 36, and 3.4.4 = 48; and 'the ratio 4:3 joined
to 5 ' does not mean either the product or the sum of 3, 4, 5, but simply
the triangle 3, 4, 5.
THE GEOMETRICAL NUMBER 307
irrational diameters, by two, and on the other hand of 100
cubes of 3.'
The ratio 4 : 3 must be taken in the sense of ' the numbers
4 and 3 ', and Adam takes 'joined with 5 ' to mean that 4, 3
and 5 are multiplied together, making 60 ; 60 ' thrice increased '
he interprets as '60 thrice multiplied by 60 ', that is to say,
60 x 60 x 60 x 60 or 3600 2 ; ' so many times 100 ' must then
be the 'equal' side of this, or 36 times 100; this 3600 2 , or
12960000, is one of the 'harmonies'. The other is the same
number expressed as the product of two unequal factors, an
'oblong' number; the first factor is 100 times a number
which can be described either as 1 less than the square of the
1 rational diameter of 5 ', or as 2 less than the square of
the ' irrational diameter ' of 5, where the irrational diameter
of 5 is the diameter of a square of side 5, i. e. \^50, and the
rational diameter is the nearest whole number to this, namely
7, so that the number which is multiplied by 100 is 49 — 1, or
50 — 2, i.e. 48, and the first factor is therefore 4800; the
second factor is 100 cubes of 3, or 2700; and of course
4800 x 2700 == 3600 2 or 12960000. Hultsch obtains the side,
3600, of the first ' harmony ' in another way ; he takes 4 and 3
joined to 5 to be the sum of 4, 3 and 5, i. e. 12, and rph avgrjOeis,
' thrice increased ', to mean that the 1 2 is ' multiplied by three' 1
making 36 ; ' so many times 100 ' is then 36 times 100, or 3600.
But the main interest of the passage from the historical
1 Adam maintains that rp\s av^Beis cannot mean ' multiplied by 3 \ He
observes (p. 278, note) that the Greek for ' multiplied by 3 ', if we
use au£aVo), would be rpidbi nv^rjOeis, this being the construction used by
Nicomuchus (ii. 15. 2 Xva 6 6 rpls y a>v ndXiv rpidSt en SXXo bidarrjpn
av^rjdfj ku\ yevrjrm o k() and in Theol. Ar. (p. 39, Ast <?£aSi nv^rjdds). Never-
theless I think that Tpls avt-rjdeis would not be an unnatural expression for
a mathematician to use for ' multiplied by 3 ', let alone Plato in a passage
like this. It is to be noted that 7ro\Xa/r\«o-ia£a> and noXXa7rXdaios are
likewise commonly used with the dative of the multiplier; yet lo-anis
noWnnXdo-ios is the regular expression for ' equimultiple '. And avj-dva is
actually found with Toaravrdias : see Pappus ii, p. 28. 15, 22, where roaav-
raKis av^aofxeu means ' we have to multiply by such a power' of 10000 or
of 10 (although it is true that the chapter in which the expression occurs
may be a late addition to Pappus's original text). On the whole, I prefer
Hultsch's interpretation to Adam's, rpls avj-rjOtU can hardly mean that
60 is raised to the fourth power, 60 4 ; and if it did, ' so many times 100 ',
immediately following the expression for 3600 2 , would be pointless and
awkward. On the other hand, 'so many times 100' following the ex-
pression for 36 would naturally indicate 3600.
x 2
308 PLATO
point of view lies in the terms ' rational ' and ' irrational
diameter of 5 '. A fair approximation to V2 was obtained
by selecting a square number such that, if 2 be multiplied by
it, the product is nearly a square ; 25 is such a square number,
since 25 times 2, or 50, only differs by 1 from 7 2 ; conse-
quently | is an approximation to \/2. It may have been
arrived at in the tentative way here indicated ; we cannot
doubt that it was current in Plato's time ; nay, we know that
the general solution of the equations
x 2 ~2y 2 = ± 1
by means of successive ' side- ' and ' diameter- ' numbers was
Pythagorean, and Plato was therefore, here as in so many
other places, ' Pythagorizing '.
The diameter is again mentioned in the Politicus, where
Plato speaks of ' the diameter which is in square (Svpdfxei)
two feet', meaning the diagonal of the square with side
1 foot, and again of the diameter of the square on this
diameter, i. e. the diagonal of a square 2 square feet in area,
in other words, the side of a square 4 square feet in area,
or a straight line 2 feet in length. 1
Enough has been said to show that Plato was abreast of
the mathematics of his day, and we can understand the
remark of Proclus on the influence which he exerted upon
students and workers in that field :
1 he caused mathematics in general and geometry in particular
to make a very great advance by reason of his enthusiasm
for them, which of course is obvious from the way in which
he filled his books with mathematical illustrations and every-
where tries to kindle admiration for these subjects in those
who make a pursuit of philosophy.' 2
Mathematical i arts '.
Besides the purely theoretical subjects, Plato recognizes the
practical or applied mathematical ' arts ' ; along with arith-
metic, he mentions the art of measurement (for purposes of
trade or craftsmanship) and that of weighing 3 ; in the former
connexion he speaks of the instruments of the craftsman,
the circle-drawer (ropvos), the compasses (Siaf3rJTr]s), the rule
1 Politicus, 266 b. 2 Proclus on Eucl. I, p. 66. 8-14.
3 Philebus, 55 E-56 e.
MATHEMATICAL 'ARTS' 309
((TTddfLT]) and ' a certain elaborate irpocray^yLov ' (? approxi-
mator). The art of weighing, lie says, 1 ' is concerned with
the heavier and lighter weight ', as ' logistic ' deals with odd
and even in their relation to one another, and geometry with
magnitudes greater and less or equal ; in the Protagoras he
speaks of the man skilled in weighing
* who puts together first the pleasant, and second the painful
things, and adjusts the near and the far on the balance ' 2 ;
the principle of the lever was therefore known to Plato, who
was doubtless acquainted with the work of Archytas, the
reputed founder of the science of mechanics. 3
(a) Optics,
In the physical portion of the Timaeus Plato gives his
explanation of the working of the sense organs. The account
of the process of vision and the relation of vision to the
light of day is interesting, 4 and at the end of it is a reference
to the properties of mirrors, which is perhaps the first indica-
tion of a science of optics. When, says Plato, we see a thing
in a mirror, the fire belonging to the face combines about the
bright surface of the mirror with the fire in the visual current ;
the right portion of the face appears as the left in the image
seen, and vice versa, because it is the mutually opposite parts
of the visual current and of the object seen which come into
contact, contrary to the usual mode of impact. (That is, if you
imagine your reflection in the mirror to be another person
looking at you, his left eye is the image of your right, and the
left side of his left eye is the image of the right side of your
right.) But, on the other hand, the right side really becomes
the right side and the left the left when the light in com-
bination with that with which it combines is transferred from
one side to the other; this happens when the smooth part
of the mirror is higher at the sides than in the middle (i. e. the
mirror is a hollow cylindrical mirror held with its axis
vertical) , and so diverts the right portion of the visual current
to the left and vice versa. And if you turn the mirror so that
its axis is horizontal, eveiything appears upside down.
1 Charmides, 166 b. 2 Protagoras, 356 b.
*, 3 Diog. L. viii. 83. 4 Timaeus, 45 b-46 c.
310 PLATO
(P) Music.
In music Plato had the advantage of. the researches of
Archytas and the Pythagorean school into the numerical
relations of tones. In the Timaeus we find an elaborate
filling up of intervals by the interposition of arithmetic and
harmonic means l ; Plato is also clear that higher and lower
pitch are due to the more or less rapid motion of the air. 2
In like manner the different notes in the ' harmony of the
spheres ', poetically turned into Sirens sitting on each of the
eight whorls of the Spindle and each uttering a single sound,
a single musical note, correspond to the different speeds of
the eight circles, that of the fixed stars and those of the sun,
the moon, and the five planets respectively. 3
(y) Astronomy.
This brings us to Plato's astronomy. His views are stated
in their most complete and final form in the Timaeus, though
account has to be taken of other dialogues, the Phaedo, the
Republic, and the Laws. He based himself upon the early
Pythagorean system (that of Pythagoras, as distinct from
that of his successors, who were the first to abandon the
geocentric system and made the earth, with the sun, the
moon and the other planets, revolve in circles about the ' cen-
tral fire ') ; while of course he would take account of the
results of the more and more exact observations made up
to his own time. According to Plato, the universe has the
most perfect of all shapes, that of a sphere. In the centre
of this sphere rests the earth, immovable and kept there by
the equilibrium of symmetry as it were (' for a thing in
equilibrium in the middle of any uniform substance will not
have cause to incline more or less in any direction ' 4 ). The
axis of the sphere of the universe passes through the centre of
the earth, which is also spherical, and the sphere revolves
uniformly about the axis in the direction from east to west.
The fixed stars are therefore carried round in small circles
of the sphere. The sun, the moon and the five planets are
also carried round in the motion of the outer sphere, but they
have independent circular movements of their ow^n in addition.
1 Timaeus, 35 c-36 B. 2 lb. 67 b.
3 Republic, 617 b. 4 Phaedo, 109 a.
ASTRONOMY 311
These latter movements take place in a plane which cuts
at an angle the equator of the heavenly sphere ; the several
orbits are parts of what Plato calls the ' circle of the Other ',
as distinguished from the ' circle of the Same ', which is the
daily revolution of the heavenly sphere as a whole and which,
carrying the circle of the Other and the seven movements
therein along with it, has the mastery over them. The result
of the combination o'f the two movements in the case of any
one planet is to twist its actual path in space into a spiral l ;
the spiral is of course included between two planes parallel to
that of the equator at a distance equal to the maximum
deviation of the planet in its course from the equator on
either side. The speeds with which the sun, the moon and
the five planets describe their own orbits (independently
of the daily rotation) are in the following order ; the moon is
the quickest; the sun is the next quickest and Venus and
Mercury travel in company with it, each of the three taking
about a year to describe its orbit ; the next in speed is Mars,
the next Jupiter, and the last and slowest is Saturn ; the
speeds are of course angular speeds, not linear. The order
of distances from the earth is, beginning with the nearest,
as follows : moon, sun, Venus, Mercury, Mars, Jupiter, Saturn.
In the Republic all these heavenly bodies describe their own
orbits in a sense opposite to that of the daily rotation, i. e. in
the direction from west to east ; this is what we should
expect; but in the Timaeus we are distinctly told, in one
place, that the seven circles move ' in opposite senses to one
another', 2 and, in another place, that Venus and Mercury
have ' the contrary tendency ' to the sun. 3 This peculiar
phrase has not been satisfactorily interpreted. The two state-
ments taken together in their literal sense appear to imply
that Plato actually regarded Venus and Mercury as describing
their orbits the contrary way to the sun, incredible as this
may appear (for on this" hypothesis the angles of divergence
between the two planets and the sun would be capable of any
value up to 180°, whereas observation shows that they are
never far from the sun). Proclus and others refer to attempts
to explain the passages by means of the theory of epicycles ;
Chalcidius in particular indicates that the sun's motion on its
1 Timaeus, 38 E-39 b. 2 lb. 36 d. 3 lb. 38 d.
312 PLATO
epicycle (which is from east to west) is in the contrary sense
to the motion of Venus and Mercury on their epicycles
respectively (which is from west to east) l ; and this would
be a satisfactory explanation if Plato could be supposed to
have been acquainted with the theory of epicycles. But the
probabilities are entirely against the latter supposition. All,
therefore, that can be said seems to be this. Heraclides of
Pontus, Plato's famous pupil, is known on clear evidence to
have discovered that Venus and Mercury revolve round the
sun like satellites. He may have come to the same conclusion
about the superior planets, but this is not certain ; and in any
case he must have made the discovery with reference to
Mercury and Venus first. Heraclides's discovery meant that
Venus and Mercury, while accompanying the sun in its annual
motion, described what are really epicycles about it. Now
discoveries of this sort are not made without some preliminary
seeking, and it may have been some vague inkling of the
truth that prompted the remark of Plato, whatever the precise
meaning of the words.
The differences between the angular speeds of the planets
account for the overtakings of one planet by another, and
the combination of their independent motions with that of the
daily rotation causes one planet to appear to be overtaking
another when it is really being overtaken by it and vice
versa. 2 The sun, moon and planets are instruments for
measuring time. 3 Even the earth is an instrument for making
night and day by virtue of its not rotating about its axis,
while the rotation of the fixed stars carrying the sun with
it is completed once in twenty-four hours ; a month has passed
when the moon after completing her own orbit overtakes the
sun (the ' month ' being therefore the synodic month), and
a year when the sun has completed its own circle. According
to Plato the time of revolution of the other planets (except
Venus and Mercury, which have the same speed as the sun)
had not been exactly calculated ; nevertheless the Perfect
Year is completed ' when the relative speeds of all the eight
revolutions [the seven independent revolutions and the daily
rotation] accomplish their course together and reach their
1 Chalcidius on Timaens, cc. 81, 109, 112. 2 Thnaeus, 39 A.
3 lb. 41 e, 42 d.
ASTRONOMY 313
starting-point \* There was apparently a tradition that the
Great Year of Plato was 36000 years : this corresponds to
the minimum estimate of the precession of the equinoxes
quoted by Ptolemy from Hipparchus's treatise on the length
of the year, namely at least one-hundredth of a degree in
a year, or 1° in 100 years, 2 that is to say, 360° in 36000 years.
The period is connected by Adam with the Geometrical Num-
ber 1296000O because this number of days, at the rate of 360
days in the year, makes 36000 years. The coincidence may,
it is true, have struck Ptolemy and made him describe the
Great Year arrived at on the basis of 1° per 100 years
as the ' Platonic ' year ; but there is nothing to show that
Plato himself calculated a Great Year with reference to pre-
cession : on the contrary, precession was first discovered by
Hipparchus.
As regards the distances of the sun, moon and planets
Plato has nothing more definite than that the seven circles
are ' in the proportion of the double intervals, three of each ' 3 :
the reference is to the Pythagorean rerpaKrv? represented in
the annexed figure, the numbers after 1 being
on the one side successive powers of 2, and on
the other side successive powers of 3. This
gives 1, 2, 3, 4, 8, 9, 27 in ascending order.
What precise estimate of relative distances
Plato based upon these figures is uncertain.
It is generally supposed (1) that the radii of the successive
orbits are in the ratio of the numbers; but (2) Chalcidius
considered that 2, 3, 4 ... are the successive differences
between these radii, 4 so that the radii themselves are in
the ratios of 1, 1+2=3, 1 + 2 + 3 = 6, &c. ; and again (3),
according to Macrobius, 5 the Platonists held that the successive
radii are as 1, 1 . 2 = 2, 1 . 2 . 3 = 6, 6 . 4 = 24, 24 . 9 = 216,
216.8 = 1728 and 1728 . 27 = 46656. In any case the
figures have no basis in observation.
We have said that Plato made the earth occupy the centre
of the universe and gave it no movement of any kind. Other
1 Timaeus, 39 b-d.
2 Ptolemy, Syntaxis, vii. 2, vol. ii, p. 15. 9-17, Heib.
3 Timaeus, 36 D. 4 Chalcidius on Timaeus, c. 96, p. 167, Wrobel
5 Macrobius, In somn. Scip. ii. 3. 14.
314 PLATO
views, however; have been attributed to Plato by later writers.
In the Timaeus Plato had used of the earth the expression
which has usually been translated ' our nurse, globed (IXXo-
fievr]v) round the axis stretched from pole to pole through
the universe '- 1 It is well known that Aristotle refers to the
passage in these terms :
1 Some say that the earth, actually lying at the centre (kou
KfUfjLtvqv km rod Kevrpov), is yet wound and mcfoes (iXXtaOai
kcli KLvtlo-6ou) about the axis stretched through the universe
from pole to pole.' 2
This naturally implies that Aristotle attributed to Plato
the view that the earth rotates about its axis. Such a view
is, however, entirely inconsistent with the whole system
described in the Timaeus (and also in the Laivs, which Plato
did not live to finish), where it is the sphere of the fixed
stars which by its revolution about the earth in 24 hours
makes night and day ; moreover, there is no reason to doubt
the evidence that it was Heraclides of Pontus who was the
first to affirm the rotation of the earth about its own axis
in 24 hours. The natural inference seems to be that Aristotle
either misunderstood or misrepresented Plato, the ambiguity
of the word IXXouevqv 'being the contributing cause or the
pretext as the case may be. There are, however, those who
maintain that Aristotle must have known what Plato meant
and was incapable of misrepresenting him on a subject like
this. Among these is Professor Burnet, 3 who, being satisfied
that Aristotle understood tXXouerrji/ to mean motion of some
sort, and on the strength of a new reading which he has
adopted from two MSS. of the first class, has essayed a new
interpretation of Plato's phrase. The new reading differs
from the former texts in having the article rrjv after
iXXofitvqv, which makes the phrase run thus, yr\v Se rpoqbbv
fj.€u rj/xerepav, IXXo/iej/rju Se Tr\v irepl rbv Sloc ttclvtos ttoXov
Tera\ievov. Burnet, holding that we can only supply with
ttjv some word like 6S6v, understands nepioSov or nepicpopdv,
and translates ' earth our nurse going to and fro on its path
round the axis which stretches right through the universe'.
' TifHtt€ilS 40 B.
2 Arist. De caelo, ii. 13, 293 b 20; cf. ii. 14, 296 a 25.
3 Greek Philosophy, Fart I, Thales to Plato, pp. 347-8.
ASTRONOMY 315
In confirmation of this Burnet cites the ' unimpeachable
testimony ' of Theophrastus, who said that
' Plato in his old age repented of having given the earth
the central place in the universe, to which it had no right ' x ;
and he concludes that, according to Plato in the Timaeus,
the earth is not the centre of the universe. But the sentences
in which Aristotle paraphrases the tWofiivrjv in the Timaeus
by the words iXXeaOai kou KiveTaOaL both make it clear that
the persons who held the view in question also declared
that the earth lies or is 'placed at the centre (Keifiivrju km
tov KevTpov), or ' placed the earth at the centre ' (kirl rod ueaov
Sevres). Burnet's explanation is therefore in contradiction to
part of Aristotle's statement, if not to the rest ; so that he
does not appear to have brought the question much nearer
to a solution. Perhaps some one will suggest that the rotation
or oscillation about the axis of the universe is small, so small
as to be fairly consistent with the statement that the earth
remains at the centre. Better, I think, admit that, on our
present information, the puzzle is insoluble.
The dictum of Theophrastus that Plato in his old age
repented of having placed the earth in the centre is incon-
sistent with the theory of the Timaeus, as we have said.
Boeckh explained it as a misapprehension. There appear
to have been among Plato's immediate successors some who
altered Plato's system in a Pythagorean sense and who may
be alluded to in another passage of the Be caelo 2 ; Boeckh
suggested, therefore, that the views of these Pythagorizing
Platonists may have been put down to Plato himself. But
the tendency now seems to be to accept the testimony of
Theophrastus literally. Heiberg does so, and so does Burnet,
who thinks it probable that Theophrastus heard the statement
which he attributes to Plato from Plato himself. But I would
point out that, if the Timaeus, as Burnet contends, contained
Plato's explicit recantation of his former view that the earth
was at the centre, there was no need to supplement it by an
oral communication to Theophrastus. In any case the question
has no particular importance in comparison with the develop-
ments which have next to be described.
1 Plutarch, Quaest. Plat. 8. 1, 1006 c ; 'cf. Life ofNuma, c. II.
2 . Arist. De caelo, ii. 13, 293 a 27-b 1.
X
FROM PLATO TO EUCLID
Whatevek original work Plato himself did in mathematics
(and it may not have been much), there is no doubt that his
enthusiasm for the subject in all branches and the pre-eminent
place which he gave it in his system had enormous influence
upon its development in his lifetime and the period following.
In astronomy we are told that Plato set it as a problem to
all earnest students to find ' what are the uniform and ordered
movements by the assumption of which the apparent move-
ments of the planets can be accounted for ' ; our authority for
this is Sosigenes, who had it from Eudemus. 1 One answer
to this, representing an advance second to none in the history
of astronomy, was given by Heraclides of Pontus, one of
Plato's pupils (circa 388-310 B.C.); the other, which was
by Eudoxus and on purely mathematical lines, constitutes
one of the most remarkable achievements in pure geometry
that the whole of the history of mathematics can show.
Both were philosophers of extraordinary range. Heraclides
wrote works of the highest class both in matter and style :
the catalogue of them covers subjects ethical, grammatical,
musical and poetical, rhetorical, historical ; and there were
geometrical and dialectical treatises as well. Similarly
Eudoxus, celebrated as philosopher, geometer, astronomer,
geographer, physician and legislator, commanded and enriched
almost the whole field of learning.
Heraclides of Pontus : astronomical discoveries.
Heraclides held that the apparent daily revolution of the
heavenly bodies round the earth was accounted for, not by
1 Simpl. on Be caelo, ii. 12 (292 b 10), p. 488. 20-34, Heib.
HERACLIDES. ASTRONOMICAL DISCOVERIES 317
the circular motion of the stars round the earth, but by the
rotation of the earth about its own axis ; several passages
attest this, e. g.
'Heraclides of Pontus supposed that the earth is in the
centre and rotates (lit. ' moves in a circle ') while the heaven
is at rest, and he thought by this supposition to save the
phenomena.' 1
True, Heraclides may not have been alone in holding this
view, for we are told that Ecphantus of Syracuse, a Pytha-
gorean, also asserted that ' the earth, being in the centre
of the universe, moves about its own centre in an eastward
direction ' 2 ; when Cicero 3 says the same thing of Hicetas, also
of Syracuse, this is probably due to a confusion. But there
is no doubt of the originality of the other capital discovery
made by Heraclides, namely that Venus and Mercury revolve,
like satellites, round the sun as centre. If, as Schiaparelli
argued, Heraclides also came to the same conclusion about
Mars, Jupiter and Saturn, he anticipated the hypothesis of
Tycho JBrahe (or rather improved on it), but the evidence is
insufficient to establish this, and I think the probabilities are
against it; there is some reason for thinking that it was
Apollonius of Perga who thus completed what Heraclides had
begun and put forward the full Tychonic hypothesis. 4 But
there is nothing to detract from the merit of Heraclides in
having pointed the way to it.
Eudoxus's theory of concentric spheres is even more re-
markable as a mathematical achievement ; it is worthy of the
man who invented the great theory of proportion set out
in Euclid, Book V, and the powerful method of exhaustion
which not only enabled the areas of circles and the volumes
of pyramids, cones, spheres, &c, to be obtained, but is at the
root of all Archimedes's further developments in the mensura-
tion of plane and solid figures. But, before we come to
Eudoxus, there are certain other names to be mentioned.
1 Simpl. on Be caelo, p. 519. 9-11, Heib. ; cf. pp. 441. 31-445. 5, pp. 541.
27-542. 2 ; Proclus in Tim. 281 E.
2 Hippolytus, Refut. i. 15 (Vors. i 3 , p. 340. 31), cf. Aetius, hi. 13. 3
(Vors.i 3 , p. 341.8-10).
3 Cic. Acad. Pr. ii. 39, 123.
4 Aristarchus of Samos, the ancient Copernicus, ch. xviii.
318 FROM PLATO TO EUCLID
Theory of numbers (Speusippus, Xenocrates).
To begin with arithmetic or the theory of numbers. Speu-
sippus, nephew of Plato, who succeeded him as head of the
school, is said to have made a particular study of Pythagorean
doctrines, especially of the works of Philolaus, and to have
written a small treatise On the Pythagorean Numbers of
which a fragment, mentioned above (pp. 72, 75, 76) is pre-
served in the Theologumena Arithmetices. 1 To judge by the
fragment, the work was not one of importance. The arith-
metic in it was evidently of the geometrical type (polygonal
numbers, for example, being represented by dots making up
the particular figures). The portion of the book dealing with
1 the five figures (the regular solids) which are assigned to the
cosmic elements, their particularity and their community
with one another ', can hardly have gone beyond the putting
together of the figures by faces, as we find it in the Timaeus.
To Plato's distinction of the fundamental triangles, the equi-
lateral, the isosceles right-angled, and the half of an equilateral
triangle cut off by a perpendicular from a vertex on the
opposite side, he adds a distinction (' passablement futile ',
as is the whole fragment in Tannery's opinion) of four
pyramids (1) the regular pyramid, with an equilateral triangle
for base and all the edges equal, (2) the pyramid on a square
base, and (evidently) having its four edges terminating at the
corners of the base equal, (3) the pyramid which is the half of
the preceding one obtained by drawing a plane through the
vertex so as to cut the base perpendicularly in a diagonal
of the base, (4) a pyramid constructed on the half of an
equilateral triangle as base ; the object was, by calling these
pyramids a monad, a dyad, a triad and a tetrad respectively,
to make up the number 10, the special properties and virtues
of which as set forth by the Pythagoreans were the subject of
the second half of the work. Proclus quotes a few opinions
of Speusippus; e.g., in the matter of theorems and problems,
he differed from Menaechmus, since he regarded both alike
as being more properly theorems, while Menaechmus would
call both alike problems?
1 Theol. Ar., Ast, p. 61.
2 Proclus on Eucl. I, pp. 77. 16 ; 78. 14.
THEORY OF NUMBERS 319
Xenocrates of Chalcedon (396-314 b. c), who succeeded
Speusippus as head of the school, having been elected by
a majority of only a few votes over Heraclides, is also said
to have written a book On Numbers and a Theory of Numbers,
besides books on geometry. 1 These books have not survived,
but we learn that Xenocrates upheld the Platonic tradition in
requiring of those who would enter the school a knowledge of
music, geometry and astronomy ; to one who was not pro-
ficient in these things he said * Go thy way, for thou hast not
the means of getting a grip of philosophy '. Plutarch says
that he put at 1,002,000,000,000 the number of syllables which
could be formed out of the letters of the alphabet. 2 If the
story is true, it represents the first attempt on record to solve
a difficult problem in permutations and combinations. Xeno-
crates was a supporter of ' indivisible lines '(and magnitudes)
by which he thought to get over the paradoxical arguments
of Zeno. 3
The Elements. Proclus's summary (continued).
In geometry we have more names mentioned in the sum-
mary of Proclus. 4
' Younger than Leodamas were Neoclides and his pupil Leon,
who added many things to what was known before their
time, so that Leon was actually able to make a collection
of the elements more carefully designed in respect both of
the number of propositions proved and of their utility, besides
which he invented diorismi (the object of which is to deter-
mine) when the problem under investigation is possible of
solution and when impossible.'
Of Neoclides and Leon we know nothing more than what
is here stated ; but the definite recognition of the Siopur/jLos,
that is, of the necessity of finding, as a preliminary to the
solution of a problem, the conditions for the possibility of
a solution, represents an advance in the philosophy and
technology of mathematics. Not that the thing itself had
not been met with before : there is, as we have seen, a
1 Diog. L. iv. 13, 14.
2 Plutarch, Quaest. Conviv. viii. 9. 13, 733 a.
3 Simpl. in Phys., p. 138. 3, &c.
4 Proclus on Eucl. I, p. 66. 18-67. 1.
320 FROM PLATO TO EUCLID
Siopicr/tos indicated in the famous geometrical passage of the
Meno 1 ; no doubt, too, the geometrical solution by the Pytha-
goreans of the quadratic equation would incidentally make
clear to them the limits of possibility corresponding to the
8iopio-fj.6s in the solution of the most general form of quad-
ratic in Eucl. VI. 27-9, where, in the case of the 'deficient'
parallelogram (Prop. 28), the enunciation states that ' the
given rectilineal figure must not be greater than the parallelo-
gram described on half of the straight line and similar to the
defect \ Again, the condition of the possibility of constructing
a triangle out of three given straight lines (Eucl. I. 22),
namely that any two of them must be together greater than
the third, must have been perfectly familiar long before Leon
or Plato.
Proclus continues : 2
' Eudoxus of Cnidos, a little younger than Leon, who had
been associated with the school of Plato, was the first to
increase the number of the so-called general theorems ; he
also added three other proportions to the three already known,
and multiplied the theorems which originated with Plato
about the section, applying to them the method of analysis.
Amyclas [more correctly Amyntas] of Heraclea, one of the
friends of Plato, Menaechmus, a pupil of Eudoxus who had
also studied with Plato, and Dinostratus, his brother, made
the whole of geometry still more perfect. Theudius of
Magnesia had the reputation of excelling in mathematics as
well as in the other branches of philosophy ; for he put
together the elements admirably and made many partial (or
limited) theorems more general. Again, Athenaeus of Cyzicus,
who lived about the same time, became famous in other
branches of mathematics and most of all in geometry. These
men consorted together in the Academy and conducted their
investigations in common. Hermotimus of Colophon carried
further the investigations already opened up by Eudoxus and
Theaetetus, discovered many propositions of the Elements
and compiled some portion of the theory of Loci. Philippus
of Meclma, who was a pupil of Plato and took up mathematics
at his instance, not only carried out his investigations in
accordance with Plato's instructions but also set himself to
do whatever in his view contributed to the philosophy of
Plato.'
1 Plato, Meno, 87 a. 2 Proclus on Eucl. I , p. 67. 2-68. 4.
THE ELEMENTS 321
It will be well to dispose of the smaller names in this
list before taking up Eudoxus, the principal subject of
this chapter. The name of Amyclas should apparently be
Amyntas, 1 although Diogenes Laertius mentions Amyclos of
Heraclea in Pontus as a pupil of Plato 2 and has elsewhere an
improbable story of one Amyclas, a Pythagorean, who with
Clinias is supposed to have dissuaded Plato from burning the
works of Democritus in view of the fact that there were
many other copies in circulation. 3 Nothing more is known
of Amyntas, Theudius, Athenaeus and Hermotimus than what
is stated in the above passage of Proclus. It is probable,
however, that the propositions, &c, in elementary geometry
which are quoted by Aristotle were taken from the Elements
of Theudius, which would no doubt be the text-book of the
time just preceding Euclid. Of Menaechmus and Dinostratus
we have already learnt that the former discovered conic
sections, and used them for finding two mean proportionals,
and that the latter applied the quadratrix to the squaring
of the circle. Philippus of Medma (vulg. Mende) is doubtless
the same person as Philippus of Opus, who is said to have
revised and published the Laws of Plato which had been left
unfinished, and to have been the author of the Epinomis.
He wrote upon astronomy chiefly ; the astronomy in the
Epinomis follows that of the Latvs and the Timaeus ; but
Suidas records the titles of other works by him as follows :
On the distance of the sun and moon, On the eclipse of the
moon, On the size of the sun, the moon and the earth, On
the planets. A passage of Aetius 4 and another of Plutarch 5
alluding to his proofs about the shape of the moon may
indicate that Philippus was the first to establish the complete
theory of the phases of the moon. In mathematics, accord-
ing to the same notice by Suidas, he wrote Arithmetica,
Means, On polygonal numbers, Cyclica, Optics, Enoptrica
(On mirrors) ; but nothing is known of the contents of these
works.
1 See hid. Hercul., ed. B cheler, Ind. Schol. Gryphisw., 1869/70, col.
6 in.
2 Diog. L. iii. 46. 3 lb. ix. 40.
4 Dox. Gr., p. 360.
5 Non posse suaviter vivi secundum Epicurum, c. 11, 1093 E.
1623 Y
322 FROM PLATO TO EUCLID
According to Apollodorus, Eudoxus flourished in 01. 103 =
368-365 B. c, from which we infer that he was born about 408
B.C., and (since he lived 53 years) died about 355 B.C. In his
23rd year he went to Athens with the physician Theomedon,
and there for two months he attended lectures on philosophy
and oratory, and in particular the lectures of Plato ; so poor
was he that he took up his abode at the Piraeus and trudged
to Athens and back on foot each day. It would appear that
his journey to Italy and Sicily to study geometry with
Archytas, and medicine with Philistion, must have been
earlier than the first visit to Athens, for from Athens he
returned to Cnidos, after which he went to Egypt with
a letter of introduction to King Nectanebus, given him by
Agesilaus ; the date of this journey was probably 381-380 B.C.
or a little later, and he stayed in Egypt sixteen months.
After that he went to Cyzicus, where he collected round him
a large school which he took with him to Athens in 368 B.C.
or a little later. There is apparently no foundation for the
story mentioned by Diogenes Laertius that he took up a hostile
attitude to Plato, 1 nor on the other side for the statements
that he went with Plato to Egypt and spent thirteen years
in the company of the Egyptian priests, or that he visited
Plato when Plato was with the younger Dionysius on his
third visit to Sicily in 361 B. c Returning later to his native
place, Eudoxus was by a popular vote entrusted with legisla-
tive office.
When in Egypt Eudoxus assimilated the astronomical
kriowledge of the priests of Heliopolis and himself made
observations. The observatory between Heliopolis and Cerce-
sura used by him was still pointed out in Augustus's time ;
he also had one built at Cnidos, and from" there he observed
the star Canopus which was not then visible in higher
latitudes. It was doubtless to record the observations thus
made that he wrote the two books attributed to him by
Hipparchus, the Mirror and the Phaenomena 2 ; it seems, how-
ever, unlikely that there could have been two independent
works dealing with the same subject, and the latter, from which
1 Diog. L. viii. 87.
2 Hipparchus, in Arati et Eudoxi i)haenomena commentarii, i. 2. 2, p. 8.
15-20 Manitius.
EUDOXUS 323
the poem of Aratus was drawn, so far as verses 19-732 are
concerned, may have been a revision of the former work and
even, perhaps, posthumous.
But it is the theoretical side of Eudoxus's astronomy rather
than the observational that has importance for us; and,
indeed, no more ingenious and attractive hypothesis than
that of Eudoxus's system of concentric spheres has. ever been
put forward to account for the apparent motions of the sun,
moon and planets. It was the first attempt at a purely
mathematical theory of astronomy, and, with the great and
immortal contributions which he made to geometry, puts him
in the very first rank of mathematicians of all time. He
was a man of science if there ever was one. N© occult or
superstitious lore appealed to him ; Cicero says that Eudoxus,
' in astrologia iudicio doctissimorum hominum facile princeps ',
expressed the opinion and left it on record that no sort of
credence should be given to the Chaldaeans in their predic-
tions and their foretelling of the life of individuals from the
day of their birth. 1 Nor would he indulge in vain physical
speculations on things which were inaccessible to observation
and experience in his time ; thus, instead of guessing at
the nature of the sun, he said that he would gladly be
burnt up like Phaethon if at that price he could get to the
sun and so ascertain its form, size, and nature. 2 Another
story (this time presumably apocryphal) is to the effect
that he grew old at the top of a very high mountain in
the attempt to discover the movements of the stars and the
heavens. 3
In our account of his work we will begin with the sentence
about him in Proclus's summary. First, he is said to have
increased ' the number of the so-called general theorems '.
'So-called general theorems' is an odd phrase; it occurred to
me whether this could mean theorems which were true of
everything falling under the conception of magnitude, as are
the definitions and theorems forming part of Eudoxus's own
theory of proportion, which applies to numbers, geometrical
magnitudes of all sorts, times, &c. A number of propositions
1 Cic, De div. ii. 42.
2 Plutarch, Non posse suaviter vivi secundum. Epicuruni, c. 11, 1094 B.
3 Petronius Arbiter, Satyr icon, 88.
Y 2
324 FROM PLATO TO EUCLID
at the beginning of Euclid's Book X similarly refer to magni-
tudes in general, and the proposition X. 1 or its equivalent
was actually used by Eudoxus in his method of exhaustion,
as it is by Euclid in his application of the same method to the
theorem (among others) of XII. 2 that circles are to one
another as the squares on their diameters.
The three ' proportions ' or means added to the three pre-
viously known (the arithmetic, geometric and harmonic) have
already been mentioned (p. 86), and, as they are alterna-
tively attributed to others, they need not detain us here.
Thirdly, we are told that Eudoxus ' extended ' or ' increased
the number of the (propositions) about the section (to, we pi
tt)v ro\ir]v) which originated with Plato, applying to them
the method of analysis '. What is the section 1 The sugges-
tion which has been received with most favour is that of
Bretschneider, 1 who pointed out that up to Plato's time there
was only one ' section ' that had any real significance in
geometry, namely the section of a straight line in extreme
and mean ratio which is" obtained in Eucl. II. 1 1 and is used
again in Eucl. IV. 10-14 for the construction of a pentagon..
These theorems were, as we have seen, pretty certainly Pytha-
gorean, like the whole of the substance of Euclid, Book II.
Plato may therefore, says Bretschneider, have directed atten-
tion afresh to this subject and investigated the metrical rela-
tions between the segments of a straight line so cut, while
Eudoxus may have continued the investigation where Plato
left off. Now the passage of Proclus says that, in extending
the theorems about ' the section ', Eudoxus applied the method
of analysis; and we actually find in Eucl. XIII. 1-5 five
propositions about straight lines cut in extreme and mean
ratio followed, in the.MSS., by definitions of analysis and
synthesis, and alternative proofs of the same propositions
in the form of analysis followed by synthesis. Here, then,
Bretschneider thought he had found a fragment of some actual
work by Eudoxus corresponding to Proclus's description.
But it is certain that the definitions and the alternative proofs
were interpolated by some scholiast, and, judging by the
figures (which are merely straight lines) and by comparison
1 Bretschneider, Die Geometrie und 'die Geometer vor Eulrfeides, pp.
167-9.
EUDOXUS 325
with the remarks on analysis and synthesis quoted from
Heron by An-Nairizi at the beginning of his commentary on
Eucl. Book II, it seems most likely that the interpolated defini-
tions and proofs were taken from Heron. Bretschneider's
argument based on Eucl. XIII. 1-5 accordingly breaks down,
and all that can be said further is that, if Eudoxus investi-
gated the relation between the segments of the straight line,
he would find in it a case of incommensurability which would
further enforce the necessity for a theory of proportion which
should be applicable to incommensurable as well as to com-
mensurable magnitudes. Proclus actually observes that
1 theorems about sections like those in Euclid's Second Book
are common to both [arithmetic and geometry] except that in
which the straight line is cat in extreme and mean ratio ' l
(cf. Eucl. XIII. 6 for the actual proof of the irrationality
in this case). Opinion, however, has not even in recent years
been unanimous in favour of Bretschneider's interpretation ;
Tannery 2 in particular preferred the old view, which pre-
vailed before Bretschneider, that i section ' meant section of
solids, e. g. by planes, a line of investigation which would
naturally precede the discovery of conies ; he pointed out that
the use of the singular, rr]v TOfirjv, which might no doubt
be taken as ' section ' in the abstract, is no real objection, that
there is no other passage which speaks of a certain section
"par excellence , and that Proclus in the words just quoted
expresses himself quite differently, speaking of ' sections ' of
which the particular section in extreme and mean ratio is
only one. Presumably the question will never be more defi-
nitely settled unless by the discovery of new documents.
(a) Theory of proportion.
The anonymous author of a scholium to Euclid's Book V,
who is perhaps Proclus, tells us that ' some say ' that this
Book, containing the general theory of proportion which is
equally applicable to geometry, arithmetic, music and all
mathematical science,' is the discovery of Eudoxus, the teacher
of Plato'. 3 There is no reason to doubt the truth of this
1 Proclus on Eucl. I, p. 60. 16-19.
2 Tannery, ha geometrie grecque, p. 76.
8 Euclid,, ed. Heib., vol. v, p. 280.
326 FROM PLATO TO EUCLID
statement. The new theory appears to have been already
familiar to Aristotle. Moreover, the fundamental principles
show clear points of contact with those used in the method
of exhaustion, also due to Eudoxus. I refer to the definition
(Eucl. V, Def. 4) of magnitudes having a ratio to one another,
which are said to be ' such as are capable, when (sufficiently)
multiplied, of exceeding one another ' ; compare with this
Archimedes's 'lemma' by means of which he says that the
theorems about the volume of a pyramid and about circles
being to one another as the squares on their diameters were
proved, namely that ' of unequal lines, unequal surfaces, or
unequal solids, the greater exceeds the less by such a
magnitude as is capable, if added (continually) to itself, of
exceeding any magnitude of those which are comparable to
one another ', i. e. of magnitudes of the same kind as the
original magnitudes.
The essence of the new theory was that it was applicable
to incommensurable as well as commensurable quantities ;
and its importance cannot be overrated, for it enabled
geometry to go forward again, after it had received the blow
which paralysed it for the time. This was the discovery of
the irrational, at a time when geometry still depended on the
Pythagorean theory of proportion, that is, the numerical
theory which was of course applicable only to commensurables.
The discovery of incommensurables must have caused what
Tannery described as ' un veritable scandale logique ' in
geometry, inasmuch as it made inconclusive all the proofs
which had depended on the old theory of proportion. One
effect would naturally be to make geometers avoid the use
of proportions as much as possible ; they would have to use
other methods wherever they could. Euclid's Books I-IV no
doubt largely represent the result of the consequent remodel-
ling of fundamental propositions ; ^and the ingenuity of the
substitutes devised is nowhere better illustrated than in I. 44,
45, where the equality of the complements about the diagonal
of a parallelogram is used (instead of the construction, as
in Book VI, of a fourth proportional) for the purpose of
applying to a given straight line a parallelogram in a given
angle and equal to a given triangle or rectilineal area.
The greatness of the new theory itself needs no further
EUDOXUS'S THEORY OF PROPORTION 327
argument when it is remembered that the definition of equal
ratios in Eucl. V, Def. 5 corresponds exactly to the modern
theory of irrationals due to Dedekind, and that it is word for
word the same as Weierstrass's definition of equal numbers.
(/?) The method of exhaustion.
In the preface to Book I of his treatise On the Sphere and
Cylinder Archimedes attributes to Eudoxus the proof of the
theorems that the volume of a pyramid is one-third of
the volume of the prism which has the same base and equal
height, and that the volume of a cone is one-third of the
cylinder with the same base and height. In the Method he
says that these facts were discovered, though not proved
(i. e. in Archimedes's sense of the word), by Democritus,
who accordingly deserved a great part of the credit for the
theorems, but that Eudoxus was the first to supply the
scientific proof. In the preface to the Quadrature of the Para-
bola Archimedes gives further details. He says that for the
proof of the theorem that the area of a segment of a parabola
cut off by a chord is frds of the triangle on the same base and
of equal height with the segment he himself used the ' lemma '
quoted above (now known as the Axiom of Archimedes), and
he goes on :
i The earlier geometers have also used this lemma ; for it is
by the use of this lemma that they have proved the proposi-
tions (1) that circles are to one another in the duplicate ratio
of their diameters, (2) that spheres are to one another in the
triplicate ratio of their diameters, and further (3) that every
pyramid is one third part of the prism which has the same
base with the pyramid and equal height ; also (4) that every
cone is one third part of the cylinder having the same base
with the cone and equal height they proved by assuming
a certain lemma similar to that aforesaid.'
As, according to the other passage, it was Eudoxus who
first proved the last two of these theorems, it is a safe
inference that he used for this purpose the ' lemma ' in ques-
tion or its equivalent. But was he the first to use the lemma ?
This has been questioned on the ground that one of the
theorems mentioned as having been proved by ' the earlier
geometers ' in this way is the theorem that circles are to one
328 FROM PLATO TO EUCLID
another as the squares on their diameters, which proposition,
as we are told on the authority of Eudemus, was proved
(Sdgai) by Hippocrates of Chios. This suggested to Hankel
that the lemma in question must have been formulated by
Hippocrates and used in his proof. 1 But seeing that, accord-
ing to Archimedes, ' the earlier geometers ' proved by means
of the same lemma both Hippocrates's proposition, (1) above,
and the theorem (3) about the volume of a pyramid, while
the first proof of the latter was certainly given by Eudoxus,
it is simplest to suppose that it was Eudoxus who first formu-
lated the ' lemma ' and used it to prove both propositions, and
that Hippocrates's ' proof ' did not amount to a rigorous
demonstration such as would have satisfied Eudoxus or
Archimedes. Hippocrates may, for instance, have proceeded
on the lines of Antiphon's ' quadrature ', gradually exhausting
the circles and taking the limit, without clinching the proof
by the formal reduetio ad absuvdum used in the method of
exhaustion as practised later. Without therefore detracting
from the merit of Hippocrates, whose argument may have
contained the germ of the method of exhaustion, we do not
seem to have any sufficient reason to doubt that it was
Eudoxus who established this method as part of the regular
machinery of geometry.
The ' lemma ' itself, we may observe, is not found in Euclid
in precisely the form that Archimedes gives it, though it
is equivalent to Eucl. V, Def. 4 (Magnitudes are said to have
a ratio to one another which are capable, when multiplied,
of exceeding one another). When Euclid comes to prove the
propositions about the content of circles, pyramids and cones
(XII. 2, 4-7 Por., and 10), he does not use the actual lemma of
Archimedes, but another which forms Prop. 1 of Book X, to
the effect that, if there are two unequal magnitudes and from
the greater there be subtracted more than its half (or the
half itself), from the remainder more than its half (or the half),
and if this be done continually, there will be left some magni-
tude which will be less than the lesser of the given magnitudes.
This last lemma is frequently used by Archimedes himself
(notably in the second proof of the proposition about the area
1 Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter,
p. 122.
EUDOXUS. METHOD OF EXHAUSTION 329
of a parabolic segment), and it may be the ' lemma similar
to the aforesaid ' which he says was used in the case of the
cone. But the existence of the two lemmas constitutes no
real difficulty, because Archimedes's lemma (under the form
of Eucl. V, Def. 4) is in effect used by Euclid to prove X. 1.
We are not told whether Eudoxus proved the theorem that
spheres are to one another in the triplicate ratio of their
diameters. As the proof of this in Eucl. XII. 16-18 is likewise
based on X. 1 (which is used in XII. 16), it is probable enough
that this proposition, mentioned along with the others by
Archimedes, was also first proved by Eudoxus.
Eudoxus, as we have seen, is said to have solved the problem
of the two mean proportionals by means of f curved lines '.
This solution has been dealt with above (pp. 249-51).
We pass on to the
(y) Theory of concentric spheres.
This was the first attempt to account by purely geometrical
hypotheses for the apparent irregularities of the motions of
the planets ; it included similar explanations of the apparently
simpler movements of the sun and moon. The ancient
evidence of the details of the system of concentric spheres
(which Eudoxus set out in a book entitled On speeds, Ilepl
rayfov, now lost) is contained in two passages. The first is in
Aristotle's Metaphysics, where a short notice is given of the
numbers and relative positions of the spheres postulated by
Eudoxus for the sun, moon and planets respectively, the
additions which Callippus thought it necessary to make to
the numbers of those spheres, and lastly the modification
of the system which Aristotle himself considers necessary
' if the phenomena are to be produced by all the spheres
acting in combination \* A more elaborate and detailed
account of the system is contained in Simplicius's commentary
on the Be caelo of Aristotle 2 ; Simplicius quotes largely from
Sosigenes the Peripatetic (second century A. d.), observing that
Sosigenes drew from Eudemus, who dealt with the subject
in the second book of his History of Astronomy. Ideler was
1 Aristotle, Metaph. A. 8. 1073 b 17-1074 a 14.
2 Simpl. on De caelo, p. 488. 18-24, pp. 493. 4-506. 18 Heib. ; p. 498
a 45-b 3, pp. 498 b 27-503 a 33.
330 FROM PLATO TO EUCLID
the first to appreciate the elegance of the theory and to
attempt to explain its working (1828, 1830) ; E. F. Apelt, too,
gave a fairly full exposition of it in a paper of 1849. But it
was reserved for Schiaparelli to work out a complete restora-
tion of the theory and to investigate in detail the extent
to which it could be made to account for the phenomena ; his
paper has become a classic, 1 and all accounts must necessarily
follow his.
I shall here only describe the system so far as to show its
mathematical interest. I have given fuller details elsewhere. 2
Eudoxus adopted the view which prevailed from the earliest
times to the time of Kepler, that circular motion was sufficient
to account for the movements of all the heavenly bodies.
With Eudoxus this circular motion took the form of the
revolution of different spheres, each of which moves about
a diameter as axis. All the spheres were concentric, the
common centre being the centre of the earth ; hence the name
of ' homocentric ' spheres used in later times to describe the
system. The spheres were of different sizes, one inside the
other. Each planet was fixed at a point in the equator of
the sphere which carried it, the sphere revolving at uniform
speed about the diameter joining the corresponding poles ;
that is, the planet revolved uniformly in a great circle of the
sphere perpendicular to the axis of rotation. But one such
circular motion was not enough ; in order to explain the
changes in the apparent speed of the planets' motion, their
stations and retrogradations, Eudoxus had to assume a number
of such circular motions working on each planet and producing
by their combination that single apparently irregular motion
which observation shows us. He accordingly held that the
poles of the sphere carrying the planet are not fixed, but
themselves move on a greater sphere concentric with the
carrying sphere and moving about two different poles with
uniform speed. The poles of the second sphere were simi-
larly placed on a third sphere concentric with and larger
than the first and second, and moving about separate poles
1 Schiaparelli. he sfere omocentriche di Endosso, di Callippo e di Aristotele,
Milano 1875; Germ, trans, by W. Horn in Abh. zur Gesch. d. Math., i.
Heft, 1877, pp. 101-98.
2 Aristarchus of Samos, the ancient Copernicus, pp. 193-224.
THEORY OF CONCENTRIC SPHERES 331
of its own with a speed peculiar to itself. For the planets
yet a fourth sphere was required, similarly related to the
others ; for the sun and moon Eudoxus found that, by a
suitable choice of the positions of the poles and of speeds
of rotation, he could make three spheres suffice. Aristotle
and Simplicius describe the spheres in the reverse order, the
sphere carrying the planet being the last; this makes the
description easier, because we begin with the sphere represent-
ing the daily rotaton of the heavens. The spheres which
move each planet Eudoxus made quite separate from those
which move the others ; but one sphere sufficed to produce
the daily rotation of the heavens. The hypothesis was purely
mathematical; Eudoxus did not trouble himself about the
material of the spheres or their mechanical connexion.
The moon has a motion produced by three spheres; the
first or outermost moves in the same sense as the fixed stars
from east to west in 24 hours ; the second moves about an
axis perpendicular to the plane of the zodiac circle or the
ecliptic, and in the sense of the daily rotation, i. e. from
east to west; the third again moves about an axis inclined
to the axis of the second at an angle equal to the highest
latitude attained by the moon, and from west to east ;
the moon is fixed on the equator of this third sphere. The
speed of the revolution of the second sphere was very slow
(a revolution was completed in a period of 223 lunations) ;
the third sphere produced the revolution of the moon from
west to east in the draconitic or nodal month (of 27 days,
5 hours, 5 minutes, 36 seconds) round a circle inclined to
the ecliptic at an angle equal to the greatest latitude of the
moon. 1 The moon described the latter circle, while the
circle itself was carried round by the second sphere in
a retrograde sense along the ecliptic in a period of 223
lunations; and both the inner spheres were bodily carried
round by the first sphere in 24 hours in the sense of the daily
rotation. The three spheres thus produced the motion of the
moon in an orbit inclined to the ecliptic, and the retrogression
of the nodes, completed in a period of about 18J years.
1 Simplicius (and presumably Aristotle also) confused the motions of
the second and third spheres. The above account represents what
Eudoxus evidently intended.
332 FROM PLATO TO EUCLID
The system of three spheres for the sun was similar, except
that the orbit was less inclined to the ecliptic than that of the
moon, and the second sphere moved from west to east instead
of from east to west, so that the nodes moved slowly forward
in the direct order of the signs instead of backward.
But the case to which the greatest mathematical interest
attaches is that of the planets, the motion of which is pro-
duced by sets of four spheres for each. Of each set the first
and outermost produced the daily rotation in 24 hours ; the
second, the motion round the zodiac in periods which in the
case of superior planets are equal to the sidereal periods of
revolution, and for Mercury and Venus (on a geocentric
system) one year. The third sphere had its poles fixed at two
opposite points on the zodiac circle, the poles being carried
round in the motion of the second sphere ; the revolution
of the third sphere about its poles was again uniform and
was completed in the synodic period of the planet or the time
which elapsed between two successive oppositions or conjunc-
tions with the sun. The poles of the third sphere were the
same for Mercury and Venus but different for all the other
planets. On the surface of the third sphere the poles of the
fourth sphere were fixed, the axis of the latter being inclined
to that of the former at an angle which was constant for each
planet but different for the different planets. The rotation of
the fourth sphere about its axis took place in the same time
as the rotation of the third about its axis but in the opposite
sense. On the equator of the fourth sphere the planet was
fixed. Consider now the actual path of a planet subject to
the rotations of the third and fourth spheres only, leaving out
of account for the moment the first two spheres the motion of
which produces the daily rotation and the motion along the
zodiac respectively. The problem is the following. A sphere
rotates uniformly about the fixed diameter AB. P, P f are
two opposite poles on this sphere, and a second sphere con-
centric with the first rotates uniformly about the diameter
PP' in the same time as the former sphere rotates about A B,
but in the opposite direction. M is a point on the second
sphere equidistant from P, P', i. e. a point on the equator
of the second sphere. Required to find the path of the
point M. This is not difficult nowadays for any one familiar
THEORY OF CONCENTRIC SPHERES 333
with spherical trigonometry and analytical geometry ; but
Schiaparelli showed, by means of a series of seven propositions
or problems involving only elementary geometry, that it was
well within the powers of such a geometer as Eudoxus. The
path of M in space turns out in fact to be a curve like
a lemniscate or figure-of-eight described on the surface of a
sphere, namely the fixed sphere about A B as diameter. This
1 spherical lemniscate ' is roughly shown in the second figure
above. The curve is actually the intersection of the sphere
with a certain cylinder touching it internally at the double
point 0, namely a cylinder with diameter equal to AS the
sagitta (shown in the other figure) of the diameter of the
small circle on which P revolves. But the curve is also
the intersection of either the sphere or the cylinder with
a certain cone with vertex 0, axis parallel to the axis of the
cylinder (i. e. touching the circle A OB at 0) and vertical angle
equal to the ' inclination' (the angle AO'P in the first figure).
That this represents the actual result obtained by Eudoxus
himself is conclusively proved by the facts that Eudoxus
called the curve described by the planet about the zodiac
circle the hippopede or horse-fetter, and that the same term
hippopede is used by Proclus to describe the plane curve of
similar shape formed by a plane section of an anchor-ring or
tore touching the tore internally and parallel to its axis. 1
So far account has only been taken of the motion due to
the combination of the rotations of the third and fourth
1 Proclus on Eucl. I, p. 112. 5.
334 FROM PLATO TO EUCLID
spheres. But A, B, the poles of the third sphere, are carried
round the zodiac or ecliptic by the motion of the second
sphere in a time equal to the ' zodiacal ' period of the planet.
Now the axis of symmetry of the ' spherical lemniscate ' (the
arc of the great circle bisecting it longitudinally) always lies
on the ecliptic. We may therefore substitute for the third
and fourth spheres the ' lemniscate ' moving bodily round
the ecliptic. The combination of the two motions (that of the
' lemniscate ' and that of the planet on it) gives the motion of
the planet through the constellations. The motion of the
planet round the curve is an oscillatory motion, now forward in
acceleration of the motion round the ecliptic due to the motion
of the second sphere, now backward in retardation of the same
motion ; the period of the oscillation is the period of the syno-
dic revolution, and the acceleration and retardation occupy
half the period respectively. When the retardation in the
sense of longitude due to the backward oscillation is greater
than the speed of the forward motion of the lemniscate itself,
the planet will for a time have a retrograde motion, at the
beginning and end of which it will appear stationary for a little
while, when the two opposite motions balance each other.
It will be admitted that to produce the retrogradations
in this theoretical way by superimposed axial rotations of
spheres was a remarkable stroke of genius. It was no slight
geometrical achievement, for those days, to demonstrate the
effect of the hypotheses; but this is nothing in comparison
with the speculative power which enabled the man to invent
the hypothesis which would produce the effect. It was, of
course, a much greater achievement than that of Eudoxus's
teacher Archytas in finding the two mean proportionals by
means of the intersection of three surfaces in space, a tore
with internal diameter nil, a cylinder and a cone ; the problem
solved by Eudoxus was much more difficult, and yet there
is the curious resemblance between the two solutions that
Eudoxus's hippopede is actually the section of a sphere with
a cylinder touching it internally and also with a certain
cone ; the two cases together show the freedom with which
master and pupil were accustomed to work with figures in
three dimensions, and in particular with surfaces of revolution,
their intersections, &c.
THEORY OF CONCENTRIC SPHERES 335
Callippus (about 370-300 B.C.) tried to make the system of
concentric spheres suit the phenomena more exactly by adding
other spheres; he left the number of the spheres at four in
the case of Jupiter and Saturn, but added one each to the
other planets and two each in the case of the sun and moon
(making five in all). This would substitute for the hippopede
a still more complicated elongated figure, and the matter is
not one to be followed out here. Aristotle modified the system
in a mechanical sense by introducing between each planet
and the one below it reacting spheres one less in number than
those acting on the former planet, and with motions equal
and opposite to each of them, except the outermost, respec-
tively ; by neutralizing the motions of all except the outermost
sphere acting on any planet he wished to enable that outer-
most to be the outermost acting on the planet below, so that
the spheres became one connected system, each being in actual
contact with the one below and acting on it, whereas with
Eudoxus and Callippus the spheres acting on each planet
formed a separate set independent of the others. Aristotle's
modification was not an improvement, and has no mathe-
matical interest.
The works of Aristotle are of the greatest importance to
the history of mathematics and particularly of the Elements.
His date (384-322/1) comes just before that of Euclid, so
that from the differences between his statement of things
corresponding to what we find in Euclid and Euclid's own we
can draw a fair inference as to the innovations which were
due to Euclid himself. Aristotle was no doubt a competent
mathematician, though he does not seem to have specialized
in mathematics, and fortunately for us he was fond of mathe-
matical illustrations. His allusions to particular definitions,
propositions, &c, in geometry are in such a form as to suggest
that his pupils must have had at hand some text-book where
they could find the things he mentions.. The particular text-
book then in use would presumably be that which was the
immediate predecessor of Euclid's, namely the Elements of
Theudius; for Theudius is the latest of pre-Euclidean
geometers whom the summary of Proclus mentions as a com-
piler of Elements. 1
1 Proclus on Eucl. I, p. 67. 12-16.
336 FROM PLATO TO EUCLID
The mathematics in Aristotle comes under the
following heads.
(a) First principles.
On no part of the subject does Aristotle throw more light
than on the first principles as then accepted. The most
important passages dealing with this subject are in the
Posterior Analytics. 1 While he speaks generally of ' demon-
strative sciences ', his illustrations are mainly mathematical,
doubtless because they were readiest to his hand. He gives
the clearest distinctions between axioms (which are common
to all sciences), definitions, hypotheses and postulates (which
are different for different sciences .since they relate to the
subject-matter of the particular science). If we exclude from
Euclid's axioms (1) the assumption that two straight lines
cannot enclose a space, which is interpolated, and (2) the
so-called '.Parallel- Axiom ' which is the 5th Postulate, Aris-
totle's explanation of these terms fits the classification of
Euclid quite well. Aristotle calls the axioms by various
terms, ' common (things)', 'common axioms', 'common opinions',
and this seems to be the origin of ' common notions ' (koivcu
'ivvoiai), the term by which they are described in the text
of Euclid ; the particular axiom which Aristotle is most fond
of quoting is No. 3, stating that, if equals be subtracted from
equals, the remainders are equal. Aristotle does not give any
instance of a geometrical postulate. From this we may fairly
make the important inference that Euclid's Postulates are all
his own, the momentous Postulate 5 as well as Nos. 1, 2, 3
relating to constructions of lines and circles, and No. 4 that
all right angles are equal. These postulates as well as those
which Archimedes lays down at the beginning of his book
On Plane Equilibriums (e.g. that 'equal weights balance at
equal lengths, but equal weights at unequal lengths do not
balance but incline in the direction of the weight which is
at the greater length ') correspond exactly enough to Aristotle's
idea of a postulate. This is something which, e.g., the
geometer assumes (for reasons known to himself) without
demonstration (though properly a subject for demonstration)
1 Anal. Post. i. 6. 74 b 5, i. 10. 76 a 31-77 a 4.
ARISTOTLE 337
and without any assent on the part of the learner, or even
against his opinion rather than otherwise. As regards defini-
tions, Aristotle is clear that they do not assert existence or
non-existence ; they only require to be understood. The only
exception he makes is in the case of the unit or monad and
magnitude, the existence of which has to be assumed, while
the existence of everything else has to be proved ; the things
actually necessary to be assumed in geometry are points and
lines only; everything constructed out of them, e.g. triangles,
squares, tangents, and their properties, e.g. incommensura-
bility, has to be proved to exist. This again agrees sub-
stantially with Euclid's procedure. Actual construction is
with him the proof of existence. If triangles other than the
equilateral triangle constructed in I. 1 are assumed in I. 4-21,
it is only provisionally, pending the construction of a triangle
out of three straight lines in I. 22 ; the drawing and producing
of straight lines and the describing of circles is postulated
(Postulates 1-3). Another interesting statement on the
philosophical side of geometry has reference to the geometer's
hypotheses. It is untrue, says Aristotle, to assert that a
geometer's hypotheses are false because he assumes that a line
which he has drawn is a foot long when it is not, or straight
when it is not straight. The geometer bases no conclusion on
the particular line being that which he has assumed it to be ;
he argues about what it represents, the figure itself being
a mere illustration. 1
Coming now to the first definitions of Euclid, Book I, we
find that Aristotle has the equivalents of Defs. 1-3 and 5, 6.
But for a straight line he gives Plato's definition only:
whence we may fairly conclude that Euclid's definition
was his own, as also was his definition of a plane which
he adapted from that of a straight line. Some terms seem
to have been defined in Aristotle's time which Euclid leaves
undefined, e. g. KeKXaaOai, ' to be inflected ', veveiv, to ' verge '. 2
Aristotle seems to have known Eudoxus's new theory of pro-
portion, and he uses to a considerable extent the usual
1 Arist. Anal. Post. i. 10. 76 b 39-77 a 2 ; cf. Anal. Prior, i. 41. 49 b 34 sq. ;
Metaph. N. 2. 1089 a 20-5.
2 Anal. Post. i. 10. 76 b 9.
1523 Z
338 FROM PLATO TO EUCLID
terminology of proportions; he defines similar figures as
Euclid does.
(ft) Indications of proofs differing from Euclid's.
Coming to theorems, we find in Aristotle indications of
proofs differing entirely from those of Euclid. The most
remarkable case is that of the theorem of I. 5. For the
purpose of illustrating the statement that in any syllogism
one of the propositions must be affirmative and universal
he gives a proof of the proposition as follows. 1
' For let A, B be drawn [i. e. joined] to the centre.
'If then we assumed (1) that the angle AG [i.e. A + C]
is equal to the angle BD [i. e. B + D] without asserting
generally that the angles of semicircles are equal, and again
(2) that the angle G is equal to the
angle D without making the further
assumption that the two angles of all
segments are equal, and if we then
inferred, lastly, that since the whole
angles are equal, and equal angles are
subtracted from them, the angles which
remain, namely E, F, are equal, without
assuming generally that, if equals be
subtracted from equals, the remainders are equal, we should
commit a petitio principii.'
There are obvious peculiarities of notation in this extract ;
the angles are indicated by single letters, and sums of two
angles by two letters in juxtaposition (cf. BE for D + E in
the proof cited from Archytas above, p. 215). The angles
A, B are the angles at A, B of the isosceles triangle OAB t the
same angles as are afterwards spoken of as E, F. But the
differences of substance between this and Euclid's proof are
much more striking. First, it is clear that ' mixed ' angles
('angles' formed by straight lines with circular arcs) played
a much larger part in earlier text-books than they do in
Euclid, where indeed they only appear once or twice as a
survival. Secondly, it is remarkable that the equality of
the two ' angles ' of a semicircle and of the two ' angles ' of any
segment is assumed as a means of proving a proposition so
1 Anal. Prior, i. 24. 41 b 13-22.
ARISTOTLE 339
elementary as I. 5, although one would say that the assump-
tions are no more obvious than the proposition to be proved ;
indeed some kind of proof, e.g. by superposition, would
doubtless be considered necessary to justify the assumptions.
It is a natural inference that Euclid's proof of I. 5 was his
own, and it would appear that his innovations as regards
order of propositions and methods of proof began at the very
threshold of the subject.
There are two passages l in Aristotle bearing on the theory
of parallels which seem to show that the theorems of Eucl.
I. 27, 28 are pre- Euclidean ; but another passage 2 appears to
indicate that there was some vicious circle in the theory of
parallels then current, for Aristotle alludes to a petitio prin-
cipii committed by ' those who think that they draw parallels '
(or ' establish the theory of parallels ', tcls napaWrjXovs
ypafaLv), and, as I have tried to show elsewhere, 3 a note of
Philoponus makes it possible that Aristotle is criticizing a
direction-theory of parallels such as has been adopted so
often in modern text-books. It would seem, therefore, to have
been Euclid who first got rid of the petitio principii in earlier
text-books by formulating the famous Postulate 5 and basing
I. 29 upon it.
A difference of method is again indicated in regard to the
theorem of Eucl. III. 3 1 that the angle in a semicircle is right.
Two passages of Aristotle taken together 4 show that before
Euclid the proposition was proved by means of the radius
drawn to the middle point of the
arc of the semicircle. Joining the
extremity of this radius to the ex-
tremities of the diameter respec-
tively, we have two isosceles right-
angled triangles, and the two angles,
one in each triangle, which are at the middle point of the arc,
being both of them halves of right angles, make the angle in
the semicircle at that point a right angle. The proof of the
theorem must have been completed by means of the theorem
1 Anal. Post. i. 5. 74 a 13-16 ; Anal. Prior, ii. 17. 66 a 11-15.
2 Anal. Prior, ii. 16. 65 a 4.
3 See The Thirteen Books of Euclid's Elements, vol. i, pp. 191-2 (cf.
pp. 308-9).
4 Anal. Post. ii. 11. 94 a 28 ; Metaph. e. 9. 1051 a 26.
340 FROM PLATO TO EUCLID
of III. 21 that angles in the same segment are equal, a proposi-
tion which Euclid's more general proof does not need to use.
These instances are sufficient to show that Euclid was far
from taking four complete Books out of an earlier text-book
without change ; his changes began at the very beginning,
and there are probably few, if any, groups of propositions in
which he did not introduce some improvements of arrange-
ment or method.
It is unnecessary to go into further detail regarding
Euclidean theorems found in Aristotle except to note the
interesting fact that Aristotle already has the principle of
the method of exhaustion used by Eudoxus : ' If I continually
add to a finite magnitude, I shall exceed every assigned
(' defined ', wpia-fievov) magnitude, and similarly, if I subtract,
I shall fall short (of any assigned magnitude).' l
(y) Propositions not found in Euclid,
Some propositions found in Aristotle but not in Euclid
should be mentioned. (1) The exterior angles of any polygon
are together equal to four right angles 2 ; although omitted
in Euclid and supplied by Proclus, this is evidently a Pytha-
gorean proposition. (2) The locus of a point such that its
distances from two given points are in a given ratio (not
being a ratio of equality) is a circle 3 ; this is a proposition
quoted by Eutocius from Apollonius's Plane Loci, but the
proof given by Aristotle differs very little from that of
Apollonius as reproduced by Eutocius, which shows that the
proposition was fully known and a standard proof of it was in
existence before Euclid's time. (3) Of all closed lines starting
from a point, returning to it again, and including a given
area, the circumference of a circle is the shortest 4 ; this shows
that the study of isoperimetry (comparison of the perimeters
of different figures having the same area) began long before
the date of Zenodorus's treatise quoted by Pappus and Theon
of Alexandria. (4) Only two solids can fill up space, namely
the pyramid and the cube 5 ; this is the complement of the
Pythagorean statement that the only three figures which can
1 Arist. Phys. viii. 10. 266 b 2.
2 Anal. Post. i. 24. 85 b 38 ; ii. 17. 99 a 19.
3 Meteorblogica, hi. 5. 376 a 3 sq. 4 De caelo, ii. 4. 287 a 27.
5 lb. iii. 8. 306 b 7.
ARISTOTLE 341
by being put together fill up space in a plane are the equi-
lateral triangle, the square and the regular hexagon.
(8) Curves and solids knoivn to Aristotle.
There is little beyond elementary plane geometry in Aris-
totle. He has the distinction between straight and ' curved '
lines (Ka/x7rv\aL ypa/x/iai), but the only curve mentioned
specifically, besides circles, seems to be the spiral 1 ; this
term may have no more than the vague sense which it has
in the expression ' the spirals of the heaven ' 2 ; if it really
means the cylindrical helix, Aristotle does not seem to have
realized its property, for he includes it among things which
are not such that 'any part will coincide with any other
part', whereas Apollonius later proved that the cylindrical
helix has precisely this property.
In solid geometry he distinguishes clearly the three dimen-
sions belonging to ' body '/and, in addition to parallelepipedal
solids, such as cubes, he is familiar with spheres, cones and
cylinders. A sphere he defines as the figure which has all its
radii (' lines*;from the centre ') equal, 3 from which we may infer
that Euclid's definition of it as the solid generated by the revo-
lution of a semicircle about its diameter is his own (Eucl. XI,
Def. 14). Referring to a cone, he says 4 'the straight lines
thrown out from K in the form of a cone make GK as a sort
of axis (coa-nep a£ova) ', showing that the use of the word
1 axis ' was not yet quite technical ; of conic sections he does
not seem to have had any knowledge, although he must have
been contemporary with Menaechmus. When he alludes to
' two cubes being a cube he is not speaking, as one might
suppose, of the duplication of the cube, for he is saying that
no science is concerned to prove anything outside its own
subject-matter ; thus geometry is not required to prove ' that
two cubes are a cube ' 5 ; hence the sense of this expression
must be not geometrical but arithmetical, meaning that the
product of two cube numbers is also a cube number. In the
Aristotelian Problems there is a question which, although not
mathematical in intention, is perhaps the first suggestion of
1 Phijs. v. 4. 228 b 24. 2 Metaph. B. 2. 998 a 5.
3 Fhys. ii. 4. 287 a 19. 4 Meteorologica, iii. 5. 375 b 21.
5 Anal. Post. i. 7. 75 b 12.
342 FROM PLATO TO EUCLID
a certain class of investigation. If a book in the form of a
cylindrical roll is cut by a plane and then unrolled, why is it
that the cut edge appears as a straight line if the section
is parallel to the base (i. e. is a right section), but as a crooked
line if the section is obliquely inclined (to the axis). 1 The
Problems are not by Aristotle ; but, whether this one goes
back to Aristotle or not, it is unlikely that he would think of
investigating the form of the curve mathematically.
(e) The continuous and the infinite.
Much light was thrown by Aristotle on certain general
conceptions entering into mathematics such as the ' continuous '
and the ' infinite '. The continuous, he held, could not be
made up of indivisible parts ; the continuous is that in which
the boundary or limit between two consecutive parts, where
they touch, is one and the same, and which, as the name
itself implies, is kept together, which is not possible if the
extremities are two and not one. 2 The ' infinite ' or ' un-
limited ' only exists potentially, not in actuality. The infinite
is so in virtue of its endlessly changing into something else,
like day or the Olympic games, and is manifested in different
forms, e.g. in time, in Man, and in the division of magnitudes.
For, in general, the infinite consists in something new being
continually taken, that something being itself always finite
but always different. There is this distinction between the
forms above mentioned that, whereas in the case of magnitudes
what is once taken remains, in the case of time and Man it
passes or is destroyed, but the succession is unbroken. The
case of addition is in a sense the same as that of division ;
in the finite magnitude the former takes place in the converse
way to the latter ; for, as we see the finite magnitude divided
ad infinitum, so we shall find that addition gives a sum
tending to a definite limit. Thus, in the case of a finite
magnitude, you may take a definite fraction of it and add to
it continually in the same ratio ; if now the successive added
terms do not include one and the same magnitude, whatever
it is [i. e. if the successive terms diminish in geometrical
progression], you will not come to the end of the finite
magnitude, but, if the ratio is increased so that each term
1 Probl. xvi. 6. 914 a 25. 2 Phys. v. 3. 227 all; vii. 1. 231 a 24.
ARISTOTLE ON THE INFINITE 343
does include one and the same magnitude, whatever it is, you
will come to the end of the finite magnitude, for every finite
magnitude is exhausted by continually taking from it any
definite fraction whatever. In no other sense does the infinite
exist but only in the sense just mentioned, that is, potentially
and by way of diminution. 1 And in this sense you may have
potentially infinite addition, the process being, as we say, in
a manner the same as with division ad infinitum ; for in the
case of addition you will always be able to find something
outside the total for the time being, but the total will never
exceed every definite (or assigned) magnitude in the way that,
in the direction of division, the result will pass every definite
magnitude, that is, by becoming smaller than it. The infinite
therefore cannot exist, even potentially, in the sense of exceed-
ing every finite magnitude as the result of successive addition.
It follows that the correct view of the infinite is the opposite
of that commonly held ; it is not that which has nothing
outside it, but that which always has something outside it. 2
Aristotle is aware that it is essentially of physical magnitudes
that he is speaking : it is, he says, perhaps a more general
inquiry that would be necessary to determine whether the
infinite is possible in mathematics and in the domain of
thought and of things which have no magnitude. 3
' But ', he says, ' my argument does not anyhow rob
mathematicians of their study, although it denies the existence
of the infinite in the sense of actual existence as something
increased to such an extent that it cannot be gone through
(dSiegiTrjTov) ; for, as it is, they do not even need the infinite
or use it, but only require that the finite (straight line) shall
be as long as they please. . . . Hence it will make no difference
to them for the purpose of demonstration/ 4
The above disquisition about the infinite should, I think,
be interesting to mathematicians for the distinct expression
of Aristotle's view that the existence of an infinite series the
terms of which are magnitudes is impossible unless it is
convergent and (with reference to Riemann's developments)
that it does not matter to geometry if the straight line is not
'nfinite in length provided that it is as long as we please.
1 Phys. iii. 6. 206 a 15-b 13. 2 lb. iii. 6. 206 b 16-207 a 1.
3 lb. iii. 5. 204 a 34. 4 lb. iii. 7. 207 b 27.
344 FROM PLATO TO EUCLID
Aristotle's denial of even the potential existence of a sum
of magnitudes which shall exceed every definite magnitude
was, as he himself implies, inconsistent with the lemma or
assumption used by Eudoxus in his method of exhaustion.
We can, therefore, well understand why, a century later,-
Archimedes felt it necessary to justify his own use of the
lemma :
' the earlier geometers too have used this lemma : for it is by
its help that they have proved that circles have to one another
the duplicate ratio of their diameters, that spheres have to
one another the triplicate ratio of their diameters, and so on.
And, in the result, each of the said theorems has been accepted
no less than those proved without the aid of this lemma.' 1
({) Mechanics.
An account of the mathematics in Aristotle would be incom-
plete without a reference to his ideas in mechanics, where he
laid down principles which, even though partly erroneous,
held their ground till the time of Benedetti (1530-90) and
Galilei (1564-1642). The Mechanica included in the Aris-
totelian writings is not indeed Aristotle's own work, but it is
very close in date, as we may conclude from its terminology ;
this shows more general agreement with the terminology of
Euclid than is found in Aristotle's own writings, but certain
divergences from Euclid's terms are common to the latter and
to the Mechanica ; the conclusion from which is that the
Mechanica was written before Euclid had made the termino-
logy of mathematics more uniform and convenient, or, in the
alternative, that it was composed after Euclid's time by persons
who, though they had partly assimilated Euclid's terminology,
were close enough to Aristotle's date to be still influenced
by his usage. But the Aristotelian origin of many of the
ideas in the Mechanica is proved by their occurrence in
Aristotle's genuine writings. Take, for example, the principle
of the lever. In the Mechanica we are told that,
'as the weight moved is to the moving weight, so is the
length (or distance) to the length inversely. In fact the mov-
ing weight will more easily move (the system) the farther it
is away from the fulcrum. The reason is that aforesaid,
1 Archimedes, Quadrature of a Parabola, Preface.
ARISTOTELIAN MECHANICS 345
namely that the line which is farther from the centre describes
the greater circle, so that, if the power applied is the same,
that which moves (the system) will change its position the
more, the farther it is away from the fulcrum.' 1
The idea then is that the greater power exerted by the
weight at the greater distance corresponds to its greater
velocity. Compare with this the passage in the De caelo
where Aristotle is speaking of the speeds of the circles of
the stars :
1 it is not at all strange, nay it is inevitable, that the speeds of
circles should be in the proportion of their sizes.' 2 . . . ' Since
in two concentric circles the segment (sector) of the outer cut
off between two radii common to both circles is greater than
that cut off on the inner, it is reasonable that the greater circle
should be carried round in the same time.' 3
Compare again the passage of the Mechanica :
1 what happens with the balance is reduced to (the case of the)
circle, the case of the lever to that of the balance, and
practically everything concerning mechanical movements to
the case of the lever. Further it is the fact that, given
a radius of a circle, no two points of it move at the same
speed (as the radius itself revolves), but the point more distant
from the centre always moves more quickly, and this is the
reason of many remarkable facts about the movements of
circles which will appear in the sequel/ 4
The axiom which is regarded as containing the germ of the
principle of virtual velocities is enunciated, in slightly different
forms, in the De caelo and the Physics :
' A smaller and lighter weight will be given more movement
if the force acting on it is the same. . . . The speed of the
lesser body will be to that of the greater as the greater body
is to the lesser.' 5
'If A be the movent, B the thing moved, C the length
through which it is moved, D the time taken, then
A will move \B over the distance 2 C in the time D,
and A „ %B „ „ C „ „ JD;
thus proportion is maintained.' 6
1 Mechanica, 3. 850 b 1. 2 De caelo, ii. 8. 289 b 15.
5 lb. 290 a 2. 4 Mechanica, 848 a 11.
5 De caelo, iii. 2. 301 b 4, 11, 6 Phys. vii. 5'. 249 b 30-250 a 4.
346 FROM PLATO TO EUCLID
Again, says Aristotle,
A will move B over the distance \G in the time \D,
and \A „ \B a distance C „ „ B; 1
and so on.
Lastly, we have in the Mechanica the parallelogram of
velocities :
• When a body is moved in a certain ratio (i. e. has two linear
movements in a constant ratio to one another), the body must
move in a straight line, and this straight line is the diameter
of the figure (parallelogram) formed from the straight lines
which have the given ratio.' 2
The author goes on to say 3 that, if the ratio of the two
movements does not remain the same from one instant to the
next, the motion will not be in a straight line but in a curve.
He instances a circle in a vertical plane with a point moving
along it downwards from the topmost point; the point has
two simultaneous movements ; one is in a vertical line, the
other displaces this vertical line parallel to itself away from
the position in which it passes through the centre till it
reaches the position of a tangent to the circle ; if during this
time the ratio of the two movements were constant, say one of
equality, the point would not move along the circumference
at all but along the diagonal of a rectangle.
The parallelogram of forces is easily deduced from the
parallelogram of velocities combined with Aristotle's axiom
that the force which moves a given weight is directed along
the line of the weight's motion and is proportional to the
distance described by the weight in a given time.
Nor should we omit to mention the Aristotelian tract On
indivisible lines. We have seen (p. 293) that, according to
Aristotle, Plato objected to the genus 'point' as a geometrical
fiction, calling a point the beginning of a line, and often
positing 'indivisible lines' in the same sense. 4 The idea of
indivisible lines appears to have been only vaguely conceived
by Plato, but it took shape in his school, and with Xenocrates
1 Phys. vii. 5. 250 a 4-7. 2 Mechanica, 2. 848 b 10.
3 lb. 848 b 26 sq. 4 Metaph. A. 9. 992 a 20.
THE TRACT ON INDIVISIBLE LINES 347
became a definite doctrine. There is plenty of evidence for
this 1 ; Proclus, for instance, tells us of ' a discourse or argu-
ment by Xenocrates introducing indivisible lines \ 2 The tract
On indivisible lines was no doubt intended as a counterblast
to Xenocrates. It can hardly have been written by Aristotle
himself; it contains, for instance, some expressions without
parallel in Aristotle. But it is certainly the work of some
one belonging to the school ; and we can imagine that, having
on some occasion to mention ' indivisible lines ', Aristotle may
well have set to some pupil, as an exercise, the task of refuting
Xenocrates. According to Simplicius and Philoponus, the
tract was attributed by some to Theophrastus 3 ; and this
seems the most likely supposition, especially as Diogenes
Laertius mentions, in a list of works by Theophrastus, ' On
indivisible lines, one Book '. The text is in many places
corrupt, so that it is often difficult or impossible to restore the
argument. In reading the book we feel that the writer is
for the most part chopping logic rather than contributing
seriously to the philosophy of mathematics. The interest
of the work to the historian of mathematics is of the slightest.
It does indeed cite the equivalent of certain definitions and
propositions in Euclid, especially Book X (on irrationals), and
in particular it mentions the irrationals called ' binomial ' or
1 apotome ', though, as far as irrationals are concerned, the
writer may have drawn on Theaetetus rather than Euclid.
The mathematical phraseology is in many places similar to
that of Euclid, but the writer shows a tendency to hark back
to older and less fixed terminology such as is usual in
Aristotle. The tract begins with a section stating the argu-
ments for indivisible lines, which we may take to represent
Xenocrates's own arguments. The next section purports to
refute these arguments one by one, after which other con-
siderations are urged against indivisible lines. It is sought to
show that the hypothesis of indivisible lines is not reconcilable
with the principles assumed, or the conclusions proved, in
mathematics ; next, it is argued that, if a line is made up
of indivisible lines (whether an odd or even number of such
lines), or if the indivisible line has any point in it, or points
1 Cf. Zeller, ii. I 4 , p. 1017. 2 Proclus on Eucl. I, p. 279. 5.
3 See Zeller, ii. 2 3 , p. 90, note.
348 FROM PLATO TO EUCLID
terminating it, the indivisible line must be divisible ; and,
lastly, various arguments are put forward to show that a line
can no more be made up of points than of indivisible lines,
with more about the relation of points to lines, &C. 1
Sphaeric.
Autolycus of Pitane was the teacher of Arcesilaus (about
315-241/40 B.C.), also of Pitane, the founder of the so-called
Middle Academy. He may be taken to have flourished about
310 B.C. or a little earlier, so that he was an elder con-
temporary of Euclid. We hear of him in connexion with
Eudoxus's theory of concentric spheres, to which he adhered.
The great difficulty in the way of this theory was early seen,
namely the impossibility of reconciling the assumption of the
invariability of the distance of each planet with the observed
differences in the brightness, especially of Mars and Venus,
at different times, and the apparent differences in the relative
sizes of the sun and moon. We are told that no one before
Autolycus had even attempted to deal with this difficulty
' by means of hypotheses ', i. e. (presumably) in a theoretical
manner, and even he was not successful, as clearly appeared
from his controversy with Aristotherus 2 (who was the teacher
of Aratus) ; this implies that Autolycus's argument was in
a written treatise.
Two works by Autolycus have come down to us. They
both deal with the geometry of the sphere in its application
to astronomy. The definite place which they held among
Greek astronomical text-books is attested by the fact that, as
we gather from Pappus, one of them, the treatise On the
moving Sphere, was included in the list of works forming
the ' Little Astronomy ', as it was called afterwards, to distin-
guish it from the 'Great Collection' (fieydXrj <tvvtol£is;) of
Ptolemy ; and we may doubtless assume that the other work
On Risings and Settings was similarly included.
1 A revised text of the work is included in Aristotle, De plantis, edited
by 0. Apelt, who also gave a German translation of it in Beitrcige zur
Geschichte der griechischen Philosophie (1891), pp. 271-86. A translation
by H. H. Joachim has since appeared (1908) in the series of Oxford
Translations of Aristotle's works.
2 Simplicius on De caelo, p. 504. 22-5 Heib.
AUTOLYCUS OF PITANE 349
Both works have been well edited by Hultsch with Latin
translation. 1 They are of great interest for several reasons.
First, Autolycus is the earliest Greek mathematician from
whom original treatises have come down to us entire, the next
being Euclid, Aristarchus and Archimedes. That he wrote
earlier than Euclid is clear from the fact that Euclid, in his
similar work, the Phaenomena, makes use of propositions
appearing in Autolycus, though, as usual in such cases, giving
no indication of their source. The form of Autolycus's proposi-
tions is exactly the same as that with which we are familiar
in Euclid ; we have first the enunciation of the proposition in
general terms, then the particular enunciation with reference
to a figure with letters marking the various points in it, then
the demonstration, and lastly, in some cases but not in all, the
conclusion in terms similar to those of the enunciation. This
shows that Greek geometrical propositions had already taken
the form which we recognize as classical, and that Euclid did
not invent this form or introduce any material changes.
A lost text-book on Sphaeric.
More important still is the fact that Autolycus, as well as
Euclid, makes use of a number of propositions relating to the
sphere without giving any proof of them or quoting any
authority. This indicates that there was already in existence
in his time a text-book of the elementary geometry of the
sphere, the propositions of which were generally known to
mathematicians. As many of these propositions are proved
in the Sphaerica of Theodosius, a work compiled two or three
centuries later, we may assume that the lost text-book proceeded
on much the same lines as that of Theodosius, with much the
same order of propositions. Like Theodosius's Sphaerica
it treated of the stationary sphere, its sections (great and
small circles) and their properties. The geometry of the
sphere at- rest is of course prior to the consideration of the
sphere in motion, i. e. the sphere rotating about its axis, which
is the subject of Autolycus's works. Who was the author of
the lost pre-Euclidean text-book it is impossible to say ;
1 Autolyci De sphaera quae movetur liber, De ortibus et occasibus libri duo
edidit F. Hultsch (Teubner 1885).
350 FROM PLATO TO EUCLID
Tannery thought that we could hardly help attributing it to
Eudoxus. The suggestion is natural, seeing that Eudoxus
showed, in his theory of concentric spheres, an extraordinary
mastery of the geometry of the sphere ; on the other hand,
as Loria observes, it is, speaking generally, dangerous to
assume that a work of an unknown author appearing in
a certain country at a certain time must have been written
by a particular man of science simply because he is the only
man of the time of whom we can certainly say that he was
capable of writing it. 1 The works of Autolycus also serve to
confirm the pre -Euclidean origin of a number of propositions
in the Elements. Hultsch 2 examined this question in detail
in a paper of 1886. There are (1) the propositions pre-
supposed in one or other of Autolycus's theorems. We have
also to take account of (2) the propositions which would be
required to establish the propositions in sphaeric assumed by
Autolycus as known. The best clue to the propositions under
(2) is the actual course of the proofs of the corresponding
propositions in the Spthaerica of Theodosius; for Theodosius
was only a compiler, and we may with great probability
assume that, where Theodosius uses propositions from Euclid's
Elements, propositions corresponding to them were used to
prove the analogous propositions in the fourth-century
Sphaeric. The propositions which, following this criterion,
we may suppose to have been directly used for this purpose
are, roughly, those represented by Eucl. I. 4, 8, 17, 19, 26, 29,
47; III. 1-3, 7, 10, 16 Cor., 26, 28, 29; IV. 6; XI. 3, 4, 10,11,
12, 14, 16, 19, and the interpolated 38. It is, naturally, the
subject-matter of Books I, III, and XI that is drawn upon,
but, of course, the propositions mentioned by no means
exhaust the number of pre-Euclidean propositions even in
those Books. When, however, Hultsch increased the list of
propositions by adding the whole chain of propositions (in-
cluding Postulate 5) leading up to them in Euclid's arrange-
ment, he took an unsafe course, because it is clear that many
of Euclid's proofs were on different lines from those used
by his predecessors.
1 Loria, Le scienze esatte nelV antica Grecia, 1914, p. 496-7.
2 Berichte der Kgl. Sachs. Gesellschaft der Wissenschaften zu Leipzig,
Phil.-hist. Classe, 1886, pp. 128-55.
AUTOLYCUS AND EUCLID
351
The work On the moving Sphere assumes abstractly a
sphere moving about the axis stretching from pole to pole,
and different series of circular sections, the first series being
great circles passing through the poles, the second small
circles (as well as the equator) which are sections of the
sphere by planes at right angles to the axis and are called
the * parallel circles', while the third kind are great circles
inclined obliquely to the axis of the sphere; the motion of
points on these circles is then considered in relation to the
section by a fixed plane through the centre of the sphere.
It is easy to recognize in the oblique great circle in the sphere
the ecliptic or zodiac circle, and in the section made by the
fixed plane the horizon, which is described as the circle
in the sphere 'which defines (opifav) the visible and the
invisible portions of the sphere'. To give an idea of the
content of the work, I will quote a few enunciations from
Autolycus and along with two of them, for the sake of
comparison with Euclid, the corresponding enunciations from
the Phaenomena.
Autolycus.
1. If a sphere revolve uni-
formly about its own axis, all
the points on the surface of the
sphere which are not on the
axis will describe parallel
circles which have the same
poles as the sphere and are
also at right angles to the axis.
Euclid.
7. If the circle in the sphere 3. The circles which are at
defining the visible and the right angles to the axis and
invisible portions of the sphere cut the horizon make both
be obliquely inclined to the their risings and settings at
axis, the circles which are at the same points of the horizon,
right angles to the axis and cut
the defining circle [horizon]
always make both their risings
and settings at the same points
of the defining circle [horizon]
and further will also be simi-
larly inclined to that circle.
352
FROM PLATO TO EUCLID
7. That the circle of the
zodiac rises and sets over the
whole extent of the horizon
between the tropics is mani-
fest, forasmuch as it touches
circles greater than those
which the horizon touches.
Autolycus. Euclid.
9. If in a sphere a great
circle which is obliquely in-
clined to the axis define the
visible and the invisible por-
tions of the sphere, then, of
the points which rise at the
same time, those towards the
visible pole set later and, of
those which set at the same
time, those towards the visible
pole rise earlier.
11. If in a sphere a great
circle which is obliquely in-
clined to the axis define the
visible and the invisible por-
tions of the sphere, and any
other oblique great circle
touch greater (parallel) circles
than those which the defin-
ing circle (horizon) touches,
the said other oblique circle
makes its risings and settings
over the whole extent of the
circumference (arc) of the de-
fining circle included between
the parallel circles which it
touches.
It will be noticed that Autolycus's propositions are more
abstract in so far as the ' other oblique circle ' in Autolycus
is any other oblique circle, whereas in .Euclid it definitely
becomes the zodiac circle. In Euclid ' the great circle defining
the visible and the invisible portions of the sphere ' is already
shortened into the technical term ' horizon ' (6pi£coi>), which is
defined as if for the first time : ' Let the name horizon be
given to the plane ■ through us (as observers) passing through
the universe and separating off the hemisphere which' is visible
above the earth.'
The book On Risings and Settings is of astronomical interest
only, and belongs to the region of Phaenomena as understood
by Eudoxus and Aratus, that is, observational astronomy.
It begins with definitions distinguishing between ' true ' and
AUTOLYCUS ON RISINGS AND SETTINGS 353
' apparent ' morning- and evening-risings and settings of fixed
stars. The ' true ' morning-rising (setting) is when the star
rises (sets) at the moment of the sun's rising; the 'true'
morning-rising (setting) is, therefore invisible to us, and so is
the ' true ' evening-rising (setting) which takes place at the
moment when the sun is setting. The 'apparent' morning-
rising (setting) takes place when the star is first seen rising
(setting) before the sun rises, and the ' apparent ' evening-
rising (setting) when the star is last seen rising (setting) after
the sun has set. The following are the enunciations of a few
of the propositions in the treatise.
I. 1. In the case of each of the fixed stars the apparent
morning-risings and settings are later than the true, and
the apparent evening-risings and settings are earlier than
the true.
I. 2. Each of the fixed stars is seen rising each night from
the (time of its) apparent morning-rising to the time of its
apparent evening-rising but at no other period, and the time
during which the star is seen rising is less than half a year.
I. 5. In the case of those of the fixed stars which are on the
zodiac circle, the interval from the time of their apparent
evening-rising to the time of their apparent evening-setting is
half a year, in the case of those north of the zodiac circle
more than half a year, and in the case of those south of the
zodiac circle less than half a year.
II. 1. The twelfth part of the zodiac circle in which the
sun is, is neither seen rising nor setting, but is hidden ; and
similarly the twelfth part which is opposite to it is neither
seen setting nor rising but is visible above the earth the whole
of the nights.
II. 4. Of the fixed stars those which are cut oft* by the
zodiac circle in the northerly or the southerly direction will
reach their evening-setting at an interval of five months from
their morning-rising.
II. 9. Of the stars which are carried on the same (parallel-)
circle those which are cut off by the zodiac circle in the
northerly direction will be hidden a shorter time than those
on the southern side of the zodiac.
1523 A a
XI
EUCLID
Date and traditions.
We have very few particulars of the lives of the great
mathematicians of Greece. Even Euclid is no exception.
Practically all that is known about him is contained in a few
sentences of Proclus's summary :
' Not much younger than these (sc. Hermotimus of Colophon
and Philippus of Mende or Medma) is Euclid, who put to-
gether the Elements, collecting many of Eudoxus's theorems,
perfecting many of Theaetetus's, and also bringing to irre-
fragable demonstration the things which were only somewhat
loosely proved by his predecessors. This man lived in the
time of the first Ptolemy. For Archimedes, who came
immediately after the first (Ptolemy), makes mention of
Euclid ; and further they say that Ptolemy once asked him if
there was in geometry any shorter way than that of the
Elements, and he replied that there was no royal road to
geometry. He is then younger than the pupils of Plato, but
older than Eratosthenes and Archimedes, the latter having
been contemporaries, as Eratosthenes somewhere says.' 1
This passage shows that even Proclus had no direct know-
ledge of Euclid's birthplace, or of the dates of his birth and
death ; he can only infer generally at what period he flourished.
All that is certain is that Euclid was later than the first
pupils of Plato and earlier than Archimedes. As Plato died
in 347 B.C. and Archimedes lived from 287 to 212 B.C., Euclid
must have flourished about 300 B.C., a date which agrees well
with the statement that he lived under the first Ptolemy, who
reigned from 306 to 283 B.C.
1 Proclus on Eucl. I, p. 68. 6-20.
DATE AND TRADITIONS 355
More particulars are, it is true, furnished by Arabian
authors. We are told that
' Euclid, son of Naucrates, and grandson of Zenarchus [the
Fihrist has ' son of Naucrates, the son of Berenice (?) '], called
the author of geometry, a philosopher of somewhat ancient
date, a Greek by nationality, domiciled at Damascus, born at
Tyre, most learned in the science of geometry, published
a most excellent and most useful work entitled the foundation
or elements of geometry, a subject in which no more general
treatise existed before among the Greeks : nay, there was no
one even of later date who did not walk in his footsteps and
frankly profess his doctrine. Hence also Greek, Roman,
and Arabian geometers not a few, who undertook the task of
illustrating this work, published commentaries, scholia, and
notes upon it, and made an abridgement of the work itself.
For this reason the Greek philosophers used to post up on the
doors of their schools the well-known notice, " Let no one
come to our school, who has not first learnt the elements
of Euclid".' 1
This shows the usual tendency of the Arabs to romance.
They were in the habit of recording the names of grand-
fathers, while the Greeks were not ; Damascus and Tyre were
no doubt brought in to gratify the desire which the Arabians
always showed to connect famous Greeks in some way or other
with the east (thus they described Pythagoras as a pupil of the
wise Salomo, and Hipparchus as ' the Chaldaean '). We recog-
nize the inscription over the doors of the schools of the Greek
philosophers as a variation of Plato's fxrjSeh dyeco/ieTp^To?
elaiTco; the philosopher has become Greek philosophers in
general, the school their schools, while geometry has become
the Elements of Euclid. The Arabs even explained that the
name of Euclid, which they pronounced variously as Uclides or
Icludes, was compounded of Ucli, a key, and Bis, a measure, or,
as some say, geometry, so that Uclides is equivalent to the
key of geometry !
In the Middle Ages most translators and editors spoke of
Euclid as Euclid of Megara, confusing our Euclid with Euclid
the philosopher, and the contemporary of Plato, who lived about
400 B.C. The first trace of the confusion appears in Valerius
1 Casiri, Biblioiheca Arabico-Hispana Escurialensis, i, p. 339 (Casiri's
source is the Tarikh al-Hukama of al-Qifti (d. 1248).
A a 2
356 EUCLID
Maximus (in the time of Tiberius) who says 1 that Plato,
on being appealed to for a solution of the problem of doubling
the cube, sent the inquirers to ' Euclid the geometer '. The
mistake was seen by one Constantinus Lascaris (d. about
1493), and the first translator to point it out clearly was
Commandinus (in his translation of Euclid published in 1572).
Euclid may have been a Platonist, as Proclus says, though
this is not certain. In any case, he probably received his
mathematical training in Athens from the pupils of Plato ;
most of the geometers who could have taught him were of
that school. But he himself taught and founded a school
at Alexandria, as we learn from Pappus's statement that
Apollonius ' spent a very long time with the pupils of Euclid
*at Alexandria'. 2 Here again come in our picturesque
Arabians, 3 who made out that the Elements were originally
written by a man whose name was Apollonius, a carpenter,
who wrote the work in fifteen books or sections (this idea
seems to be based on some misunderstanding of Hypsicles's
preface to the so-called Book XIV of Euclid), and that, as
some of the work was lost in course of time and the rest
disarranged, one of the kings at Alexandria who desired to
study geometry and to master this treatise in particular first
questioned about it certain learned men who visited him, and
then sent for Euclid, who was at that time famous as a
geometer, and asked him to revise and complete the work
and reduce it to order, upon which Euclid rewrote the work
in thirteen books, thereafter known by his name.
On the character of Euclid Pappus has a remark which,
however, was probably influenced by his obvious animus
against Apollonius, whose preface to the Conies seemed to him
to give too little credit to Euclid for his earlier work in the same
subject. Pappus contrasts Euclid's attitude to his predecessors.
Euclid, he says, was no such boaster or controversialist : thus
he regarded Aristaeus as deserving credit for the discoveries
he had made in conies, and made no attempt to anticipate
him or to construct afresh the same system, such was his
scrupulous fairness and his exemplary kindliness to all who
1 viii. 12, ext. 1. 2 Pappus, vii, p. 678. 10-12.
3 The authorities are al-Kindi, l)e institute libri Euclidis and a commen-
tary by Qadizade on the Ashkal at-ta'sis of Ashraf Shamsaddln as-Samar-
qandi (quoted by Casiri and Haji Khalfa).
DATE AND TRADITIONS 357
could advance mathematical science to however small an
extent. 1 Although, as I have indicated, Pappus's motive was
rather to represent Apollonius in a relatively unfavourable
light than to state a historical fact about Euclid, the state-
ment accords well with what we should gather from Euclid's
own works. These show no sign of any claim to be original ;
in the Elements, for instance, although it is clear that he
made great changes, altering the arrangement of whole Books,
redistributing propositions between them, and inventing new
proofs where the new order made the earlier proofs inappli-
cable, it is safe to say that he made no more alterations than
his own acumen and the latest special investigations (such as
Eudoxus's theory of proportion) showed to be imperative in
order to make the exposition of the whole subject more
scientific than the earlier efforts of writers of elements. His
respect for tradition is seen in his retention of some things
which were out of date and useless, e. g. certain definitions
never afterwards used, the solitary references to the angle
of a semicircle or the angle of a segment, and the like ; he
wrote no sort of preface to his work (would that he had !)
such as those in which Archimedes and Apollonius introduced
their treatises and distinguished what they claimed as new in
them from what was already known : he plunges at once into
his subject, ' A point is that which has no part ' !
And what a teacher he must have been ! One story enables
us to picture him in that capacity. According to Stobaeus,
' some one who had begun to read geometry with Euclid,
when he had learnt the first theorem, asked Euclid, " what
shall I get by learning these things ? " Euclid called his slave
and said, " Give him threepence, since he must make gain out
of what he learns ".' 2
Ancient commentaries, criticisms, and references.
Euclid has, of course, always been known almost exclusively
as the author of the Elements. From Archimedes onwards
the Greeks commonly spoke of him as 6 cttoi)(€1(ot7i?, the
writer of the Elements, instead of using his name. This
wonderful book, with all its imperfections, which indeed are
slight enough when account is taken of the date at which
1 Pappus, vii, pp. 676. 25-678. 6. 2 Stobaeus, Floril. iv. p. 205.
358 EUCLID
it appeared, is and will doubtless remain the greatest mathe-
matical text-book of all time. Scarcely any other book
except the Bible can have circulated more widely the world
over, or been more edited and studied. Even in Greek times
the most accomplished mathematicians occupied themselves
with it ; Heron, Pappus, Porphyry, Proclus and Simplicius
wrote commentaries ; Theon of Alexandria re-edited it, alter-
ing the language here and there, mostly with a view to
greater clearness and consistency, and interpolating inter-
mediate steps, alternative proofs, separate 'cases', porisms
(corollaries) and lemmas (the most important addition being
the second part of VI. 33 relating to sectors). Even the great
Apollonius was moved by Euclid's work to discuss the first
principles of geometry ; his treatise on the subject was in
fact a criticism of Euclid, and none too successful at that ;
some alternative definitions given by him have point, but his
alternative solutions of some of the easy problems in Book I
do not constitute any improvement, and his attempt to prove
the axioms (if one may judge by the case quoted by Proclus,
that of Axiom 1) was thoroughly misconceived.
Apart from systematic commentaries on the whole work or
substantial parts of it, there were already in ancient times
discussions and controversies on special subjects dealt with by
Euclid, and particularly his theory of parallels. The fifth
Postulate was a great stumbling-block. We know from
Aristotle that up to his time the theory of parallels had not-
been put on a scientific basis l : there was apparently some
petitio principii lurking in it. It seems therefore clear that
Euclid was the first to apply the bold remedy of laying down
the indispensable principle of the theory in the form of an
indemonstrable Postulate. But geometers were not satisfied
with this solution. Posidonius and Geminus tried to get
over the difficulty by substituting an equidistance theory of
parallels. Ptolemy actually tried to prove Euclid's postulate,
as also did Proclus, and (according to Simplicius) one Diodorus,
as well as ' Aganis ' ; the attempt of Ptolemy is given by
Proclus along with his own, while that of 'Aganis' is repro-
duced from Simplicius by the Arabian commentator an-
Nairizl.
1 Anal. Prior, ii. 16. 65 a 4.
COMMENTARIES, CRITICISMS & REFERENCES 359
Other very early criticisms there were, directed against the
very first steps in Euclid's work. Thus Zeno of Sidon, an
Epicurean, attacked the proposition I. 1 on the ground that it
is not conclusive unless it be first assumed that neither two
straight lines nor two circumferences can have a common
segment ; and this was so far regarded as a serious criticism
that Posidonius wrote a whole book to controvert Zeno. 1
Again, there is the criticism of the Epicureans that I. 20,
proving that any two sides in a triangle are together greater
than the third, is evident even to ait- ass and requires no
proof. I mention these isolated criticisms to show that the
Elements, although they superseded all other Elements and
never in ancient times had any rival, were not even at the
first accepted without question.
The first Latin author to mention Euclid is Cicero ; but
it is not likely that the Elements had then been translated
into Latin. Theoretical geometry did not appeal to the
Romans, who only cared for so much of it as was useful for
measurements and calculations. Philosophers studied Euclid,
but probably in the original Greek ; Martianus Capella speaks
of the effect of the mention of the proposition ' how to con-
struct an equilateral triangle on a given straight line ' among
a company of philosophers, who, recognizing the first pro-
position of the Elements, straightway break out into encomiums
on Euclid. 2 Beyond a fragment in a Verona palimpsest of
a free rendering or rearrangement of some propositions from
Books XII and XIII dating apparently from the fourth century,
we have no trace of any Latin version before Boetius (born
about A. d. 480), to whom Magnus Aurelius Cassiodorus and
Theodoric attribute a translation of Euclid. The so-called
geometry of Boetius which has come down to us is by no
means a translation of Euclid ; but even the redaction of this
in two Books which was edited by Friedlein is not genuine,
having apparently been put together in the eleventh century
from various sources ; it contains the definitions of Book I,
the Postulates (five in number), the Axioms (three only), then
some definitions from Eucl. II, III, IV, followed by the
enunciations only (without proofs) of Eucl. I, ten propositions
1 Proclus on Eucl. I, p. 200. 2. 2 Mart. Capella, vi. 724.
360 EUCLID
of Book II, and a few of Books III and IV, and lastly a
passage indicating that the editor will now give something of
his own, which turns out to be a literal translation of the
proofs of Eucl. I. 1-3. This proves that the Pseudo-Boetius
had a Latin translation of Euclid from which he extracted
these proofs ; moreover, the text of the definitions from
Book I shows traces of perfectly correct readings which are
not found even in the Greek manuscripts of the tenth century,
but which appear in Proclus and other ancient sources.
Fragments of such a Latin translation are also found in
the Gromatici veteres. 1
The text of the Elements.
All our Greek texts of the Elements up to a century ago
depended upon manuscripts containing Theon's recension of the
work ; these manuscripts purport, in their titles, to be either
' from the edition of Theon ' (e/c rfjs Oecovos eKSocrecos) or
'from the lectures of Theon' (dnb vvvovo-iSav rov Qeodvos).
Sir Henry Savile in his Praelectiones had drawn attention
to the passage in Theon's Commentary on Ptolemy 2 quoting
the second part of VI. 33 about sectors as having been proved
by himself in his edition of the Elements', but it was not
till Peyrard discovered in the Vatican the great MS.
gr. 190, containing neither the words from the titles of the
other manuscripts quoted above nor the addition to VI. 33,
that scholars could get back from Theon's text to what thus
represents, on the face of it, a more ancient edition than
Theon's. It is also clear that the copyist of P (as the manu-
script is called after Peyrard), or rather of its archetype,
had before him the two recensions and systematically gave
the preference to the earlier one ; for at XIII. 6 in P the first
hand has a marginal note, ' This theorem is not given in most
copies of the neiv edition, but is found in those of the old '.
The editio prlnceps (Basel, 1533) edited by Simon Grynaeus
was based on two manuscripts (Venetus Marcianus 301 and
Paris, gr. 2343) of the sixteenth century, which are among
the worst. The Basel edition was again the foundation
of the text of Gregory (Oxford, 1703), who only consulted the
1 Ed. Lachmann, pp. 377 sqq. 2 I, p. 201, ed. Halma.
THE TEXT OF THE ELEMENTS 361
manuscripts bequeathed by Savile to the University in
places where the Basel text differed from the Latin version
of Commandinus which he followed in the main. It was
a pity that even Peyrard in his edition (1814-18) only
corrected the Basel text by means of P, instead of rejecting
it altogether and starting afresh ; but he adopted many of the
readings of P and gave a conspectus of them in an appendix.
E. F. August's edition (1826-9) followed P more closely, and
he consulted the Viennese MS. gr. 103 also; but it was
left for Heiberg to bring out a new and definitive Greek text
(1883-8) based on P and the best of the Theonine manuscripts,
and taking account of external sources such as Heron and
Proclus. Except in a few passages, Proclus's manuscript does
not seem to have been of the best, but authors earlier than
Theon, e. g. Heron, generally agree with our best manuscripts.
Heiberg concludes that the Elements were most spoiled by
interpolations about the third century, since Sextus Empiricus
had a correct text, while Iamblicus had an interpolated one.
The differences between the inferior Theonine manuscripts
and the best sources are perhaps best illustrated by the arrange-
ment of postulates and axioms in Book I. Our ordinary
editions based on Simson have three postulates and twelve
axioms. Of these twelve axioms the eleventh (stating that
all right angles are equal) is, in the genuine text, the fourth
Postulate, and the twelfth Axiom (the Parallel-Postulate) is
the fifth Postulate; the Postulates were thus originally five
in number. Of the ten remaining Axioms or Common
Notions Heron only recognized the first three, and Proclus
only these and two others (that things which coincide are
equal, and that the whole is greater than the part); it is fairly
certain, therefore, that the rest are interpolated, including the
assumption that two straight lines cannot enclose a space
(Euclid himself regarded this last fact as involved in Postu-
late 1, which implies that a straight line joining one point
to another is unique).
Latin and Arabic translations.
The first Latin translations which we possess in a complete
form were made not from the Greek but from the Arabic.
It was as early as the eighth century that the Elements found
362 EUCLID
their way to Arabia. The Caliph al-Mansur (754-75), as the
result of a mission to the Byzantine Emperor, obtained a copy
of Euclid among other Greek books, and the Caliph al-Ma'mun
(813-33) similarly obtained manuscripts of Euclid, among
others, from the Byzantines. Al-Hajjaj b. Yusuf b. Matar made
two versions of the Elements, the first in the reign of Harun
ar-Rashid (786-809), the second for al-Ma'mun; six Books of
the second of these versions survive in a Leyden manuscript
(Cod. Leidensis 399. 1) which is being edited along with
an-Nairizi's commentary by Besthorn and Heiberg l ; this
edition was abridged, with corrections and explanations, but
without change of substance, from the earlier version, which
appears to be lost. The work was next translated by Abu
Ya'qub Ishaq b. Hunainb. Ishaq al- c Ibadi (died 910), evidently
direct from the Greek; this translation seems itself to have
perished, but we have it as revised by Thabit b. Qurra (died
901) in two manuscripts (No. 279 of the year 1238 and No. 280
written in 1260-1) in the Bodleian Library; Books I-XIII in
these manuscripts are in the Ishaq-Thabit version, while the
non-Euclidean Books XIV, XV are in the translation of Qusta
b. Luqa al-Ba'labakkl (died about 912). Ishaq's version seems
to be a model of good translation ; the technical terms are
simply and consistently rendered, the definitions and enun-
ciations differ only in isolated cases from the Greek, and the
translator's object seems to have been only to get rid of
difficulties and unevennesses in the Greek text while at the
same time giving a faithful reproduction of it. The third
Arabic version still accessible to us is that of Nasiraddm
at-Tusi (born in 1201 at Tus in Khurasan); this, however,
is not a translation of Euclid but a rewritten version based
upon the older Arabic translations. On the whole, it appears
probable that the Arabic tradition (in spite of its omission
of lemmas and porisms, and, except in a very few cases, of
the interpolated alternative proofs) is not to be preferred
to that of the Greek manuscripts, but must be regarded as
inferior in authority.
The known Latin translations begin with that of Athelhard,
an Englishman, of Bath ; the date of it is about 1120. That
1 Parts I, i. 1893, I, ii. 1897, II, i. 1900, II, ii. 1905, III, i. 1910 (Copen-
hagen).
LATIN AND ARABIC TRANSLATIONS 363
it was made from the Arabic is clear from the occurrence
of Arabic words in it ; but Athelhard must also have had
before him a translation of (at least) the enunciations of
Euclid based ultimately upon the Greek text, a translation
going- back to the old Latin version which was the common
source of the passage in the Gromatici and ' Boetius '. But
it would appear that even before Athelhard's time some sort
of translation, or at least fragments of one, were available
even in England if one may judge by the Old English verses :
' The clerk Euclide on this wyse hit fonde
Thys craft of gemetry yn Egypte londe
Yn Egypte he tawghte hyt ful wyde,
In dyvers londe on every syde.
Mony erys afterwarde y understonde
Yer that the craft com ynto thys londe.
Thys craft com into England, as y yow say,
Yn tyme of good Kyng Adelstone's day',
which would put the introduction of Euclid into England
as far back as A. D. 924-40.
Next, Gherard of Cremona (1114—87) is said to have
translated the ' 1 5 Books of Euclid ' from the Arabic as he
undoubtedly translated an-Nairizi's commentary on Books
I— X ; this translation of the Elements was till recently
supposed to have been lost, but in 1904 A. A. Bjornbo dis-
covered in manuscripts at Paris, Boulogne-sur-Mer and Bruges
the whole, and at Rome Books X-XV, of a translation which
he gives good ground for identifying with Gherard's. This
translation has certain Greek words such as rombus, romboides,
where Athelhard keeps the Arabic terms ; it was thus clearly
independent of Athelhard's, though Gherard appears to have
had before him, in addition, an old translation of Euclid from
the Greek which Athelhard also used. Gherard's translation
is much clearer than Athelhard's ; it is neither abbreviated
nor ' edited ' in the same way as Athelhard's, but it is a word
for w r ord translation of an Arabic manuscript containing a
revised and critical edition of Thabit's version.
A third translation from the Arabic was that of Johannes
Campanus, which came some 150 years after that of Athelhard.
That Campanus's translation was not independent of Athel-
hard's is proved by the fact that, in all manuscripts and
364 EUCLID
editions, the definitions, postulates and axioms, and the 364
enunciations are word for word identical in Athelhard and
Campanus. The exact relation between the two seems #ven
yet not to have been sufficiently elucidated. Campanus may
have used Athelhard's translation and only developed the
proofs by means of another redaction of the Arabian Euclid.
Campanus's translation is the clearer and more complete,
following the Greek text more closely but still at some
distance ; the arrangement of the two is different ; in Athel-
hard the proofs regularly precede the enunciations, while
Campanus follows the usual order. How far the differences
in the proofs and the additions in each are due to the
translators themselves or go back to Arabic originals is a
moot question ; but it seems most probable that Campanus
stood to Athelhard somewhat in the relation of a commen-
tator, altering and improving his translation by means of
other Arabic originals.
The first printed editions.
Campanus's translation had the luck to be the first to be
put into print. It was published at Venice by Erhard Ratdolt
in 1482. This beautiful and very rare book was not only
the first printed edition of Euclid, but also the first printed
mathematical book of any importance. It has margins of
2\ inches and in them are placed the figures of the proposi-
tions. Ratdolt says in his dedication that, at that time,
although books by ancient and modern authors were being
printed every day in Venice, little or nothing mathematical
had appeared ; this fact he puts down to the difficulty involved
by the figures, which no one had up to that time succeeded in
printing ; he adds that after much labour he had discovered
a method by which figures could be produced as easily as
letters. Experts do not seem even yet to be agreed as to the
actual way in which the figures were made, whether they
were woodcuts or whether they were made by putting together
lines and circular arcs as letters are put together to make
words. How eagerly the opportunity of spreading geometrical
knowledge was seized upon is proved by the number of
editions which followed in the next few years. Even the
THE FIRST PRINTED EDITIONS 365
year 1482 saw two forms of the book, though they only differ
in the first sheet. Another edition came out at Ulm in 1486,
and another at Vicenza in 1491.
In 1501 G. Valla gave in his encyclopaedic work Be ex-
petendis et fugiendis rebus a number of propositions with
proofs and scholia translated from a Greek manuscript which
was once in his possession ; but Bartolomeo Zamberti (Zam-
bertus) was the first to bring out a translation from the
Greek text of the whole of the Elements, which appeared
at Venice in 1505. The most important Latin translation
is, however, that of Commandinus (1509-75), who not only
followed the Greek text more closely than his predecessors,
but added to his translation some ancient scholia as well
as good notes of his own ; this translation, which appeared
in 1572, was the foundation of most translations up to the
time of Peyrard, including that of Simson, and therefore of
all those editions, numerous in England, which gave Euclid
' chiefly after the text of Dr. Simson '.
The study of Euclid in the Middle Ages.
A word or two about the general position of geometry in
education during the Middle Ages will not be out of place in
a book for English readers, in view of the unique place which
Euclid has till recently held as a text-book in this country.
From the seventh to the tenth century the study of geometry
languished : ' We find in the whole literature of that time
hardly the slightest sign that any one had gone farther
in this department of the Quadrivium than the definitions
of a triangle, a square, a circle, or of a pyramid or cone, as
Martianus Capella and Isidorus (Hispalensis, died as Bishop
of Seville in 636) left them.' 1 (Isidorus had disposed of the
four subjects of Arithmetic, Geometry, Music and Astronomy
in four pages of his encyclopaedic work Origines or Ety-
mologiae). In the tenth century appeared a ' reparator
studiorum ' in the person of the great Gerbert, who was born
at Aurillac, in Auvergne, in the first half of the tenth century,
and after a very varied life ultimately (in 999) became Pope
Sylvester II; he died in 1003. About 967 he went on
1 Hankel, op. cit., pp. 311-12.
366 EUCLID
a journey to Spain, where he studied mathematics. In 970 he
went to Rome with Bishop Hatto of Vich (in the province of
Barcelona), and was there introduced by Pope John XIII
to the German king Otto I. To Otto, who wished to find
him a post as a teacher, he could say that ' he knew enough of
mathematics for this, but wished to improve his knowledge
of logic'. With Otto's consent he went to Reims, where he
became Scholasticus or teacher at the Cathedral School,
remaining there for about ten years, 972 to 982. As the result
of a mathematico-philosophic argument in public at Ravenna
in 980, he was appointed by Otto II to the famous monastery
at Bobbio in Lombardy, which, fortunately for him, was rich
in valuable manuscripts of all sorts. Here he found the
famous '• Codex Arcerianus ' containing fragments of the
works of the Gromatici, Frontinus, Hyginus, Balbus, Nipsus,
Epaphroditus and Vitruvius Rufus. Although these frag-
ments are not in themselves of great merit, there are things
in them which show that the authors drew upon Heron of
Alexandria, and Gerbert made the most of them. They
formed the basis of his own ' Geometry ', which may have
been written between the years 981 and 983. In writing this
book Gerbert evidently had before him Boetius's Arithmetic,
and in the course of it he mentions Pythagoras, Plato's
Timaeus, with Chalcidius's commentary thereon, and Eratos-
thenes. The geometry in the book is mostly practical ; the
theoretical part is confined to necessary preliminary matter,
definitions, &c.,and a few proofs ; the fact that the sum of the
angles of a triangle is equal to two right angles is proved in
Euclid's manner. A great part is taken up with the solution
of triangles, and with heights and distances. The Archimedean
value of 7r ( 2 T 2 -) is used in stating the area of a circle ; the
surface of a sphere is given as |y D 3 . The plan of the book
is quite different from that of Euclid, showing that Gerbert
could neither have had Euclid's Element* before him, nor,
probably, Boetius's Geometry, if that work in its genuine
form was a version of Euclid. When in a letter written
probably from Bobbio in 983 to Adalbero, Archbishop of
Reims, he speaks of his expectation of finding ' eight volumes
of Boetius on astronomy, also the most famous of figures
(presumably propositions) in geometry and other things not
STUDY OF EUCLID IN THE MIDDLE AGES 367
less admirable ', it is not clear that he actually found these
things, and it is still less certain that the geometrical matter
referred to was Boetius's Geometry.
From Gerbert's time, again, no further progress was made
until translations from the Arabic began with Athelhard and
the rest. Gherard of Cremona (died 1187), who translated
the Elements and an-Nairizi's commentary thereon, is credited
with a whole series of translations from the Arabic of Greek
authors ; they included the Data of Euclid, the Sphaerica of
Theodosius,theAS^aeWc<xof Menelaus, the Syntaxis of Ptolemy ;
besides which he translated Arabian geometrical works such
as the Liber trium fratrum, and also the algebra of Muhammad
b. Musa. One of the first results of the interest thus aroused
in Greek and Arabian mathematics was seen in the very
remarkable works of Leonardo of Pisa (Fibonacci). Leonardo
first published in 1202, and then brought out later (1228) an
improved edition of, his Liber abaci in which he gave the
whole of arithmetic and algebra as known to the Arabs, but
in a free and independent style of his own ; in like manner in
his Practica geometriae of 1220 he collected (1) all that the
Elements of Euclid and Archimedes's books on the Measure-
ment of a Circle and On the Sphere and Cylinder had taught
him about the measurement of plane figures bounded by
straight lines, solid figures bounded by planes, the circle and
the sphere respectively, (2) divisions of figures in different
proportions, wherein he based himself on Euclid's book On the
divisions of figures, but carried the subject further, (3) some
trigonometry, which he got from Ptolemy and Arabic sources
(he uses the terms sinus rectus and sinus versus) ; in the
treatment of these varied subjects he showed the same mastery
and, in places, distinct originality. We should have expected
a great general advance in the next centuries after such a
beginning, but, as Hankel says, when we look at the work of
Luca Paciuolo nearly three centuries later, we find that the
talent which Leonardo had left to the Latin world had lain
hidden in a napkin and earned no interest. As regards the
place of geometry in education during this period we have
the evidence of Roger Bacon (1214-94), though he, it
is true, seems to have taken an exaggerated view of the
incompetence of the mathematicians and teachers of his
368 EUCLID
time ; the philosophers of his day, he says, despised geo-
metry, languages, &c., declaring that they were useless ;
people in general, not finding utility in any science such as
geometry, at once recoiled, unless they were boys forced to
it by the rod, from the idea of studying it, so that they
would hardly learn as much as three or four propositions ;
the fifth proposition of Euclid was called Elefuga or fuga
miserorum. 1
As regards Euclid at the jJniversities, it may be noted that
the study of geometry seems to have been neglected at the
University of Paris. At the reformation of the University in
1336 it was only provided that no one should take a Licentiate
who had not attended lectures on some mathematical books ;
the same requirement reappears in 1452 and 1600. From the
preface to a commentary on Euclid which appeared in 1536
we learn that a candidate for the degree of M.A. had to take
a solemn oath that he had attended lectures on the first six
Books ; but it is doubtful whether for the examinations more
than Book I was necessary, seeing that the proposition I. 47
was known as Magister matheseos. At the University of
Prague (founded in 1348) mathematics were more regarded.
Candidates for the Baccalaureate had to attend lectures on
the Tractatus de Sphaera materiali, a treatise on the funda-
mental ideas of spherical astronomy, mathematical geography
and the ordinary astronomical phenomena, but without the
help of mathematical propositions, written by Johannes de
Sacrobosco (i.e. of Holy wood, in Yorkshire) in 1250, a book
which was read at all Universities for four centuries and
many times commented upon ; for the Master's degree lectures
on the first six Books of Euclid were compulsory. Euclid
was lectured upon at the Universities of Vienna (founded 1365),
Heidelberg (1386), Cologne (1388) ; at Heidelberg an oath was
required from the candidate for the Licentiate corresponding
to M.A. that he had attended lectures on some whole books and
not merely parts of several books (not necessarily, it appears,
of Euclid) ; at Vienna, the first five Books of Euclid were
required ; at Cologne, no mathematics were required for the
Baccalaureate, but the candidate for M.A. must have attended
1 Roger Bacon, Opus Tertium, cc. iv, vi.
STUDY OF EUCLID IN THE MIDDLE AGES 369
lectures on the Sphaera mundi, planetary theory, three Books
of Euclid, optics and arithmetic. At Leipzig (founded 1409),
as at Vienna and Prague, there were lectures on Euclid for
some time at all events, though Hankel says that he found no
mention of Euclid in a list of lectures given in the consecutive
years 1437-8, and Regiomontanus, when he went to Leipzig,
found no fellow-students in geometry. At Oxford, in the
middle of the fifteenth century, the first two Books of Euclid
were read, and doubtless the Cambridge course was similar.
The first English editions.
After the issue of the first printed editions of Euclid,
beginning with the translation of Campano, published by
Ratdolt, and of the editio princess of the Greek text (1533),
the study of Euclid received a great impetus, as is shown
by the number of separate editions and commentaries which
appeared in the sixteenth century. The first complete English
translation by Sir Henry Billingsley (1570) was a monumental
work of 928 pages of folio size, with a preface by John Dee,
and notes extracted from all the most important commentaries
from Proclus down to Dee himself, a magnificent tribute to
the immortal Euclid. About the same time Sir Henry Savile
began to give unpaid lectures on the Greek geometers ; those
on Euclid do not indeed extend beyond I. 8, but they are
valuable because they deal with the difficulties connected with
the preliminary matter, the definitions, &c, and the tacit
assumptions contained in the first propositions. But it was
in the period from about 1660 to 1730, during which Wallis
and Halley were Professors at Oxford, and Barrow and
Newton at Cambridge, that the study of Greek mathematics
was at its height in England. As regards Euclid in particular
Barrow's influence was doubtless very great. His Latin
version (Eudidis Elementorum Libri XV breviter demon-
strati) came out in 1655, and there were several more editions
of the same published up to 1732 ; his first English edition
appeared in 1660, and was followed by others in 1705, 1722,
1732, 1751. This brings us to Simson's edition, first published
both in Latin and English in 1756. It is presumably from
this time onwards that Euclid acquired the unique status as
1523 B b
370 EUCLID
a text-book which it maintained' till recently. I cannot help
thinking that it was Barrow's influence which contributed
most powerfully to this. We are told that Newton, when
he first bought a Euclid in 1662 or 1663, thought it ' a trifling
book ', as the propositions seemed to him obvious ; after-
wards, however, on Barrow's advice, he studied the Elements
carefully and derived, as he himself stated, much benefit
therefrom.
Technical terms connected with the classical form
of a proposition.
As the classical form of a proposition in geometry is that
which we find in Euclid, though it did not originate with
him, it is desirable, before we proceed to an analysis of the
Elements, to give some account of the technical terms used by
the Greeks in connexion with such propositions and their
proofs. We will take first the terms employed to describe the
formal divisions of a proposition.
(a) Terms for the formal divisions of a proposition.
In its completest form a proposition contained six parts,
(1) the irpoTaais, or enunciation in general terms, (2) the
€k$€(tl?, or setting-out, which states the particular data, e.g.
a given straight line AB } two given triangles ABC, DEF, and
the like, generally shown in a figure and constituting that
upon which the proposition is to operate, (3) the Siopiorfios,
definition or specification , which means the restatement of
what it is required to do or to prove in terms of the particular
data, the object being to fix our ideas, (4) the KctTaarKevT], the
construction or machinery used, which includes any additions
to the original figure by way of construction that are necessary
to enable the proof to proceed, (5) the aTr68ei£is, or the proof
itself, and (6) the crvnTrepaafia, or conclusion, which reverts to
the enunciation, and states what has been proved or done ;
the conclusion can, of course, be stated in as general terms
as the enunciation, since it does not depend on the particular
figure drawn ; that figure is only an illustration, a type of the
class of figure, and it is legitimate therefore, in stating
the conclusion, to pass from the particular to the general.
FORMAL DIVISIONS OF A PROPOSITION 371
In particular cases some of these formal divisions may be
absent, but three are always found, the enunciation, proof
and conclusion. Thus in many propositions no construction
is needed, the given figure itself sufficing for the proof ;
again, in IV. 10 (to construct an isosceles triangle with each
of the base angles double of the vertical angle) we may, in
a sense, say with Proclus l that there is neither setting-out nor
definition, for there is nothing given in the enunciation, and
we set out, not a given straight line, but any straight line AB,
while the proposition does not state (what might be said by
way of definition) that the required triangle is to have AB for
one of its equal sides.
(/3) The SLopKTfios or statement of conditions of possibility.
Sometimes to the statement of a problem there has to be
added a Siopio-jios in the more important and familiar sense of
a criterion of the conditions of possibility or, in its most
complete form, a criterion as to ' whether what is sought
is impossible or possible and how far it is practicable and in
how many ways \ 2 Both kinds of Siopio-fios begin with the
words Sei 8rj, which should be translated, in the case of the
definition, ' thus it is required (to prove or do so and so) ' and,
in the case of the criterion of possibility, ' thus it is necessary
that . . .' (not ' but it is necessary . . .'). Cf. I. 22, ' Out of
three straight lines which are equal to three given straight
lines to construct a triangle : thus it is necessary that two
of the straight lines taken together in any manner should be
greater than the remaining straight line '.
(y) Analysis, synthesis, reduction, reductio ad absurdum.
The Elements is a synthetic treatise in that it goes directly
forward the whole way, always proceeding from the known
to the unknown, from the simple and particular to the more
complex and general ; hence analysis, which reduces the
unknown or the more complex to the known, has no place
in the exposition, though it would play an important part in
the discovery of the proofs. A full account of the Greek
analysis and synthesis will come more conveniently elsewhere.
1 Proclus on Eucl. I, p. 203. 23 sq. 2 lb., p. 202. 3.
Bb2
372 EUCLID
In the meantime we may observe that, where a proposition
is worked out by analysis followed by synthesis, the analysis
comes between the definition and the construction of the
proposition ; and it should not be forgotten that reductio ad
absurdum (called in Greek r) el? to olSvvcltov a,7raycoyrj,
'reduction to the impossible', or fj 8ia tov aSvvdrov 8ei£i?
or <xtt68€i£is, 'proof r per impossibile'), a method of proof
common in Euclid as elsewhere, is a variety of analysis.
For analysis begins with reduction (dirayoayrj) of the original
proposition, which we hypothetically assume to be true, to
something simpler which we can recognize as being either
true or false ; the case where it leads to a conclusion known
to be false is the reductio ad absurdum.
(8) Case, objection, porism^ lemma.
Other terms connected with propositions are the following.
A proposition may have several cases according to the different
arrangements of points, lines, &c, in the figure that may
result from variations in the positions of the elements given ;
the word for case is tttcoo-is. The practice of the great
geometers was, as a rule, to give only one case, leaving the
others for commentators or pupils to supply for themselves.
But they were fully alive to the existence of such other
cases ; sometimes, if we may believe Proclus, they would even
give a proposition solely with a view to its use for the purpose
of proving a case of a later proposition which is actually
omitted. Thus, according to Proclus, 1 the second part of I. 5
(about the angles beyond the base) was intended to enable
the reader to meet an objection (tva-racris) that might be
raised to I. 7 as given by Euclid on the ground that it was
incomplete, since it took no account of what was given by
Proclus himself, and is now generally given in our text-books,
as the second case.
What we call a corollary was for the Greeks a porism
(7r6pi<rfjLa), i. e. something provided or ready-made, by which
was meant some result incidentally revealed in the course
of the demonstration of the main proposition under discussion,
a sort of incidental gain ' arising out of the demonstration,
1 Proclus on Eucl. I, pp. 248. 8-11 ; 263. 4-8.
TECHNICAL TERMS 373
as Proclus says. 1 The name porism was also applied to a
special kind of substantive proposition, as in Euclid's separate
work in three Books entitled Porisms (see below, pp. 431-8).
The word lemma (Xrj/jL/j,a) simply means something assumed.
Archimedes uses it of what is now known as the Axiom of
Archimedes, the principle assumed by Eudoxus and others in
the method of exhaustion; but it is more commonly used
of a subsidiary proposition requiring proof, which, however,
it is convenient to assume in the place where it is wanted
in order that the argument may not be interrupted or unduly
lengthened. Such a lemma might be proved in advance, but
the proof was often postponed till the end, the assumption
being marked as something to be afterwards proved by some
such words as cay i£fj$ SeLxOrjo-erat, ' as will be proved in due
course \
Analysis of the Elements.
Book I of the Elements necessarily begins with the essential
preliminary matter classified under the headings Definitions
(opoi), Postulates (alrrjuara) and Common Notions (koivoiI
'ivvoiou). In calling the axioms Common Notions Euclid
followed the lead of Aristotle, who uses as alternatives for
1 axioms ' the terms ' common (things) ', e common opinions '.
Many of the Definitions are open to criticism on one ground
or another. Two of them at least seem to be original, namely,
the definitions of a straight line (4) and of a plane surface (7) ;
unsatisfactory as these are, they seem to be capable of a
simple explanation. The definition of a straight line is
apparently an attempt to express, without any appeal to
sight, the sense of Plato's definition ' that of which the middle
covers the ends ' (sc. to an eye placed at one end and looking
along it) ; and the definition of a plane surface is an adaptation
of the same definition. But most of the definitions were
probably adopted from earlier text-books ; some appear to be
inserted merely out of respect for tradition, e.g. the defini-
tions of oblong, rhombus, rhomboid, which are never used
in the Elements. The definitions of various figures assume
the existence of the thing defined, e.g. the square, and the
1 lb., p. 212. 16.
374 EUCLID
different kinds of triangle under their twofold classification
(a) with reference to their sides (equilateral, isosceles and
scalene), and (b) with reference to their angles (right-angled,
obtuse-angled and acute-angled) ; such definitions are pro-
visional pending the proof of existence by means of actual con-
struction. A parallelogram is not defined ; its existence is
first proved in I. 33, and in the next proposition it is called a
' parallelogrammic area ', meaning an area contained by parallel
lines, in preparation for the use of the simple word ' parallelo-
gram ' from t. 35 onwards. The definition of a diameter
of a circle (17) includes a theorem ; for Euclid adds that 'such
a straight line also bisects the circle ', which is one of the
theorems attributed to Thales ; but this addition was really
necessary in view of the next definition (18), for, without
this explanation, Euclid would not have been justified in
describing a semi-circle as a portion of a circle cut off' by
a diameter.
More important by far are the five Postulates, for it is in
them that Euclid lays down the real principles of Euclidean
geometry ; and nothing shows more clearly his determination
to reduce his original assumptions to the very minimum.
The first three Postulates are commonly regarded as the
postulates of construction, since they assert the possibility
(1) of drawing the straight line joining two points, (2) of
producing a straight line in either direction, and (3) of describ-
ing a circle with a given centre and ' distance '. But they
imply much more than this. In Postulates 1 and 3 Euclid
postulates the existence of straight lines and circles, and
implicitly answers the objections of those who might say that,
as a matter of fact, the straight lines and circles which we
can draw are not mathematical straight lines and circles ;
Euclid may be supposed to assert that we can nevertheless
assume our straight lines and circles to be such for the purpose
of our proofs, since they are only illustrations enabling us to
imagine the real things which they imperfectly represent.
Rut, again, Postulates 1 and 2 further imply that the straight
line drawn in the first case and the produced portion of the
straight line in the second case are unique ; in other words,
Postulate 1 implies that two straight lines cannot enclose a
space, and so renders unnecessary the ' axiom ' to that effect
THE ELEMENTS. BOOK I 375
interpolated in Proposition 4, while Postulate 2 similarly im-
plies the theorem that two straight lines cannot have a
common segment, which Simson gave as a corollary to I. 11.
At first sight the Postulates 4 (that all right angles are
equal) and 5 (the Parallel-Postulate) might seem to be of
an altogether different character, since they are rather of the
nature of theorems unproved. But Postulate 5 is easily seen
to be connected with constructions, because so many con-
structions depend on the existence and use of points in which
straight lines intersect ; it is therefore absolutely necessary to
lay down some criterion by which we can judge whether two
straight lines in a figure will or will not meet if produced.
Postulate 5 serves this purpose as well as that of providing
a basis for the theory of parallel lines. Strictly speaking,
Euclid ought to have gone further and given criteria for
judging whether other pairs of lines, e.g. a straight line and
a circle, or two circles, in a particular figure will or will not
intersect one another. But this would have necessitated a
considerable series of propositions, which it would have been
difficult to frame at so early a stage, and Euclid preferred
to assume such intersections provisionally in certain cases,
e.g. in I. 1.
Postulate 4 is often classed as a theorem. But it had in any
case to be placed before Postulate 5 for the simple reason that
Postulate 5 would be no criterion at all unless right angles
were determinate magnitudes ; Postulate 4 then declares them
to be such. But this is not all. If Postulate 4 were to be
proved as a theorem, it could only be proved by applying one
pair of ' adjacent ' right angles to another pair. This method
would not be valid unless on the assumption of the invaria-
bility of figures, which would therefore have to be asserted as
an antecedent postulate. Euclid preferred to assert as a
postulate, directly, the fact that all right angles are equal ;
hence his postulate may be taken as equivalent to the prin-
ciple of the invariability of figures, or, what is the same thing,
the homogeneity of space.
For reasons which I have given above (pp. 339, 358), I think
that the great Postulate 5 is due to Euclid himself; and it
seems probable that Postulate 4 is also his, if not Postulates
1-3 as well.
376 EUCLID
Of the Common Notions there is good reason to believe
that only five (at the most) are genuine, the first three and
two others, namely * Things which coincide when applied to
one another are equal to one another ' (4), and ' The whole
is greater than the part ' (5). The objection to (4) is that
it is incontestably geometrical, and therefore, on Aristotle's
principles, should not be classed as an ' axiom ' ; it is a more
or less sufficient definition of geometrical equality, but not
a real axiom. Euclid evidently disliked the method of super-
position for proving equality, no doubt because it assumes the
possibility of motion without deformation. But he could not
dispense with it altogether. Thus in I. 4 he practically had
to choose between using the method and assuming the whole
proposition as a postulate. But he does not there quote
Common Notion 4; he says 'the base BC will coincide with
the base EF and will be equal to it '. Similarly in I. 6 he
does not quote Common Notion 5, but says ' the triangle
DEC will be equal to the triangle ACB, the less to the greater,
which is absurd '. It seems probable, therefore, that even
these two Common Notions, though apparently recognized
by Proclus, were generalizations from particular inferences
found in Euclid and were inserted after his time.
The propositions of Book I fall into three distinct groups.
The first group consists of Propositions 1-26, dealing mainly
with triangles (without the use of parallels) but also with
perpendiculars (11, 12), two intersecting straight lines (15),
and one straight line standing on another but not cutting it,
and making 'adjacent' or supplementary angles (13, 14).
Proposition 1 gives the construction of an equilateral triangle
on a given straight line as base; this is placed here not so
much on its own account as because it is at once required for
constructions (in 2, 9, 10, 11). The construction in 2 is a
direct continuation of the minimum constructions assumed
in Postulates 1-3, and enables us (as the Postulates do not) to
transfer a given length of straight line from one place to
another; it leads in 3 to the operation so often required of
cutting off from one given straight line a length equal to
another. 9 and 1 are the problems of bisecting a given angle
and a given straight line respectively, and 11 shows how
to erect a perpendicular to a given straight line from a given
THE ELEMENTS. BOOK 1 377
point on it. Construction as a means of proving existence is
in evidence in the Book, not only in 1 (the equilateral triangle)
but in 11, 12 (perpendiculars erected and let fall), and in
22 (construction of a triangle in the general case where the
lengths of the sides are given) ; 23 constructs, by means of 22,
an angle equal to a given rectilineal angle. The propositions
about triangles include the congruence-theorems (4, 8, 26) —
omitting the ' ambiguous case ' which is only taken into
account in the analogous proposition (7) of Book VI — and the
theorems (allied to 4) about two triangles in which two sides
of the one are respectively equal to two sides of the other, but
of the included angles (24) or of the bases (25) one is greater
than the other, and it is proved that the triangle in which the
included angle is greater has the greater base and vice versa.
Proposition 7, used to prove Proposition 8, is also important as
being the Book I equivalent of III. 10 (that two circles cannot
intersect in more points than two). Then we have theorems
about single triangles in 5, 6 (isosceles triangles have the
angles opposite to the equal sides equal — Thales's theorem —
and the converse), the important propositions 16 (the exterior
angle of a triangle is greater than either of the interior and
opposite angles) and its derivative 17 (any two angles of
a triangle are together less than two right angles), 18, 19
(greater angle subtended by greater side and vice versa),
20 (any two sides together greater than the third). This last
furnishes the necessary Siopta-fios, or criterion of possibility, of
the problem in 22 of constructing a triangle out of three
straight lines of given length, which problem had therefore
to come after and not before 20. 21 (proving that the two
sides of a triangle other than the base are together greater,
but include a lesser angle, than the two sides of any other
triangle on the same base but with vertex within the original
triangle) is useful for the proof of the proposition (not stated
in Euclid) that of all straight lines drawn from an external
point to a given straight line the perpendicular is the
shortest, and the nearer to the perpendicular is less than the
more remote.
The second group (27-32) includes the theory of parallels
(27-31, ending with the construction through a given point
of a parallel to a given straight line) ; and then, in 32, Euclid
378 EUCLID
proves that the sum of the three angles of a triangle is equal
to two right angles by means of a parallel to one side drawn
from the opposite vertex (cf. the slightly different Pytha-
gorean proof, p. 143).
The third group of propositions (33-48) deals generally
with parallelograms, triangles and squares with reference to
their areas. 33, 34 amount to the proof of the existence and
the property of a parallelogram, and then we are introduced
to a new conception, that of equivalent figures, or figures
equal in area though not equal in the sense of congruent :
parallelograms on the same base or on equal bases and between
the same parallels are equal in area (35, 36) ; the same is true
of triangles (37, 38), and a parallelogram on the same (or an
equal) base with a triangle and between the same parallels is
double of the triangle (41). 39 and the interpolated 40 are
partial converses of 37 and 38. The theorem 41 enables us
' to construct in a given rectilineal angle a parallelogram
equal to a given triangle' (42). Propositions 44, 45 are of
the greatest importance, being the first cases of the Pytha-
gorean method of ' application of areas ', ' to apply to a given
straight line, in a given rectilineal angle, a parallelogram
equal to a given triangle (or rectilineal figure) '. The con-
struction in 44 is remarkably ingenious, being based on that
of 42 combined with the proposition (43) proving that the
' complements of the parallelograms about the diameter ' in any
parallelogram are equal. We are thus enabled to transform
a parallelogram of any shape into another with the same
angle and of equal area but with one side of any given length,
say a unit length ; this is the geometrical equivalent of the
algebraic operation of dividing the product of two quantities
by a third. Proposition 46 constructs a square on any given
straight line as side, and is followed by the great Pythagorean
theorem of the square on the hypotenuse of a right-angled
triangle (47) and its converse (48).' The remarkably clever
proof of 47 by means of the well-known ' windmill ' figure
and the application to it of I. 41 combined with I. 4 seems to
be due to Euclid himself; it is really equivalent to a proof by
the methods of Book VI (Propositions 8, 17), and Euclid's
achievement was that of avoiding the use of proportions and
making the proof dependent upon Book I only.
THE ELEMENTS. BOOKS I-II 379
I make no apology for having dealt at some length with
Book I and, in particular, with the preliminary matter, in
view of the unique position and authority of the Elements
as an exposition of the fundamental principles of Greek
geometry, and the necessity for the historian of mathematics
of a clear understanding of their nature and full import.
It will now be possible to deal more summarily with the
other Books.
Book II is a continuation of the third section of Book I,
relating to the transformation of areas, but is specialized in
that it deals, not with parallelograms in general, but with
rectangles and squares, and makes great use of the figure
called the gnomon. The rectangle is introduced (Def. 1) as
a ' rectangular parallelogram ', which is said to be ' contained
by the two straight lines containing the right angle \ The
gnomon is defined (Def. 2) with reference to any parallelo-
gram, but the only gnomon actually used is naturally that
which belongs to a square. The whole Book constitutes an
essential part of the geometrical algebra which really, in
Greek geometry, took the place of our algebra. The first ten
propositions give the equivalent of the following algebraical
identities.
1 . a(b + c + d + . . .) = ab + ac + ad + . . . ,
2. (a + b)a + (a + b)b = {a + bf,
3. (a + b)a = ab + a 2 ,
4. (a + bf = a 2 + b 2 + 2ab,
5. ab+{*(a + b)-b}*= ii(a + b)} 2 ,
or (a + /?)(a-/?) + /3 2 = a 2 ,
6. (2a + b)b + a 2 = {a + bf,
or (a + /?)(/3-a) + a 2 -/3 2 ,
7. (a + 6) 2 + a 2 = 2(a + b)a + b 2 ,
or a 2 + /3 2 = 2a/3 + (a-£) 2 ,
8. 4(a + 6)a + & 2 = {(a + 6) + a} 2 ;
or 4a/3 + (a-/3) 2 = (a+/?) 2 ;
380 EUCLID
9. a 2 + b 2 = 2[{i(a + b)} 2 + {i(a + b)-b}*],
or ((X + /3) 2 + ((x-p) 2 =2((x 2 +(3 2 ),
10. (2a + 6) 2 + 6 2 = 2{a 2 + (a + &) 2 },
or (a + p f + (p - oc) 2 = 2 (a 2 + P 2 ).
As we have seen (pp. 151-3), Propositions 5 and 6 enable us
to solve the quadratic equations
9 29 x + y = a)
(1) ax — or — b l or u ,„>>
v ' xy — b l )
and (2) ax + x 2 = b 2 or " •
The procedure is geometrical throughout; the areas in all
the Propositions 1-8 are actually shown in the figures.
Propositions 9 and 10 were really intended to solve a problem
in numbers, that of finding any number of successive pairs
of integral numbers (' side- ' and ' diameter- ' numbers) satisfy-
ing the equations
2x 2 — y' 2 = ±1
(see p. 93, above).
Of the remaining propositions, II. 11 and II. 14 give the
geometrical equivalent of solving the quadratic equations
x 2 + ax = a 2
and x 2 = ab,
while the intervening propositions 12 and 13 prove, for any
triangle with sides a, b, c, the equivalent of the formula
a 2 = b 2 + c 2 — 2 be cos A.
It is worth noting that, while I. 47 and its converse con-
clude Book I as if that Book was designed to lead up to the
great proposition of Pythagoras, the last propositions but one
of Book II give the generalization of the same proposition
with any triangle substituted for a right-angled triangle.
The subject of Book III is the geometry of the circle,
including the relations between circles cutting or touching
each other. It begins with some definitions, which are
THE ELEMENTS. BOOKS II-III 381
generally of the same sort as those of Book I. Definition 1,
stating that equal circles are those which have their diameters
or their radii equal, might alternatively be regarded as a
postulate or a theorem ; if stated as a theorem, it could only
be proved by superposition and the congruence-axiom. It is
curious that the Greeks had no single word for radius, which
was with them ' the (straight line) from the centre ', 77 <ek tov
KkvTpov. A tangent to a circle is defined (Def. 2) as a straight
line which meets the circle but, if produced, does not cut it ;
this is provisional pending the proof in III. 16 that such lines
exist. The definitions (4, 5) of straight lines (in a circle),
i. e. chords, equally distant or more or less distant from the
centre (the test being the length of the perpendicular from
the centre on the chord) might have referred, more generally,
to the distance of any straight line from any point. The
definition (7) of the 'angle of a, segment' (the 'mixed' angle
made by the circumference with the base at either end) is
a survival from earlier text-books (cf. Props. 16, 31). The
definitions of the ' angle in a segment ' (8) and of ' similar
segments' (11) assume (provisionally pending III. 21) that the
angle in a segment is one and the same at whatever point of
the circumference it is formed. A sector (rofievs, explained by
a scholiast as o-KvroTOfxiKos rofiev?, a shoemaker's knife) is
defined (10), but there is nothing about ' similar sectors ' and
no statement that similar segments belong to similar sectors.
Of the propositions of Book III we may distinguish certain
groups. Central properties account for four propositions,
namely 1 (to find the centre of a circle), 3 (any straight line
through the centre which bisects any chord not passing
through the centre cuts it at right angles, and vice versa),
4 (two chords not passing through the centre cannot bisect
one another) and 9 (the centre is the only point from which
more than two equal straight lines can be drawn to the
circumference). Besides 3, which shows that any diameter
bisects the whole series of chords at right angles to it, three
other propositions throw light on the form of the circum-
ference of a circle, 2 (showing that it is everywhere concave
towards the centre), 7 and 8 (dealing with the varying lengths
of straight lines drawn from any point, internal or external,
to the concave or convex circumference, as the case may be,
382 EUCLID
and proving that they are of maximum or minimum length
when they pass through the centre, and that they diminish or
increase as they diverge more and more from the maximum
or minimum straight lines on either side, while the lengths of
any two which are equally inclined to them, one on each side,
are equal).
Two circles which cut or touch one another are dealt with
in 5, 6 (the two circles cannot have the same centre), 10, 13
(they cannot cut in more points than two, or touch at more
points than one), 11 and the interpolated 12 (when they touch,
the line of centres passes through the point of contact).
14,15 deal with chords (which are equal if equally distant
from the centre and vice versa, while chords more distant
from the centre are less, and chords less distant greater, and
vice versa).
16-19 are concerned with tangent properties including the
drawing of a tangent (17); it is in 16 that we have the
survival of the 'angle of a semicircle ', which is proved greater
than any acute rectilineal angle, while the ' remaining ' angle
(the ' angle ' , afterwards called KepaToeiSrjs, or ' hornlike ',
between the curve and the tangent at the point of contact)
is less than any rectilineal angle. These ' mixed ' angles,
occurring in 16 and 31, appear no more in serious Greek
geometry, though controversy about their nature went on
in the works of commentators down to Clavius, Peletarius
(Pele'tier), Vieta, Galilei and Wallis.
We now come to propositions about segments. 20 proves
that the angle at the centre is double of the angle at the
circumference, and 21 that the angles in the same segment are
all equal, which leads to the property of the quadrilateral
in a circle (22). After propositions (23, 24) on 'similar
segments ', it is proved that in equal circles equal arcs subtend
and are subtended by equal angles at the centre or circum-
ference, and equal arcs subtend and are subtended by equal
chords (26-9). 30 is the problem of bisecting a given arc,
and 31 proves that the angle in a segment is right, acute or
obtuse according as the segment is a semicircle, greater than
a semicircle or less than a semicircle. 32 proves that the
angle made by a tangent with a chord through the point
of contact is equal to the angle in the alternate segment ;
THE ELEMENTS. BOOKS III-IV 383
33, 34 are problems of constructing or cutting off a segment
containing a given angle, and 25 constructs the complete circle
when a segment of it is given.
The Book ends with three important propositions. Given
a circle and any point 0, internal (35) or external (36), then,
if any straight line through meets the circle in P, Q, the
rectangle PO . OQ is constant and, in the case where is
an external point, is equal to the square on the tangent from
to the circle. Proposition 37 is the converse of 36.
Book IV, consisting entirely of problems, again deals with
circles, but in relation to rectilineal figures inscribed or circum-
scribed to them. After definitions of these terms, Euclid
shows, in the preliminary Proposition 1, how to fit into a circle
a chord of given length, being less than the diameter. The
remaining problems are problems of inscribing or circum-
scribing rectilineal figures. The case of the triangle comes
first, and we learn how to inscribe in or circumscribe about
a circle a triangle equiangular with a given triangle (2, 3) and
to inscribe a circle in or circumscribe a circle about a given
triangle (4, 5). 6-9 are the same problems for a square, 11-
14 for a regular pentagon, and 15 (with porism) for a regular
hexagon. The porism to 15 also states that the side of the
inscribed regular hexagon is manifestly equal to the radius
of the circle. 16 shows how to inscribe in a circle a regular
polygon with fifteen angles, a problem suggested by astronomy,
since the obliquity of the, ecliptic was taken to be about 24°,
or one-fifteenth of 360°. IV. 10 is the important proposition,
required for the construction of a regular pentagon, ' to
construct an isosceles triangle such that each of the base
angles is double of the vertical angle ', which is effected by
dividing one* of the equal sides in extreme and mean ratio
(II. 11) and fitting into the circle with this side as radius
a chord equal to the greater segment ; the proof of the con-
struction depends on III. 32 and 37.
We are not surprised to learn from a scholiast that the
whole Book is ' the discovery of the Pythagoreans '} The
same scholium says that ' it is proved in this Book that
the perimeter of a circle is not triple of its diameter, as many
1 Euclid, ed. Heib., vol. v, pp. 272-3.
384 EUCLID
suppose, but greater than that (the reference is clearly to
IV. 15 Por.), and likewise that neither is the circle three-
fourths of the triangle circumscribed about it'. Were these
fallacies perhaps exposed in the lost Pseudaria of Euclid ?
Book V is devoted to the new theory of proportion,
applicable to incommensurable as well as commensurable
magnitudes, and to magnitudes of every kind (straight lines,
areas, volumes, numbers, times, &c), which was due to
Eudoxus. Greek mathematics can boast no finer discovery
than this theory, which first put on a sound footing so much
of geometry as depended on the use of proportions. How far
Eudoxus himself worked out his theory in detail is unknown ;
the scholiast who attributes the discovery of it to him says
that ' it is recognized by all ' that Book V is, as regards its
arrangement and sequence in the Elements, due to Euclid
himself. 1 The ordering of the propositions and the develop-
ment of the proofs are indeed masterly and worthy of Euclid ;
as Barrow said, ' There is nothing in the whole body of the
elements of a more subtile invention, nothing more solidly
established, and more accurately handled, than the doctrine of
proportionals'. It is a pity that, notwithstanding the pre-
eminent place which Euclid has occupied in English mathe-
matical teaching, Book V itself is little known in detail ; if it
were, there would, I think, be less tendency to seek for
substitutes; indeed, after reading some of the substitutes,
it is with relief that one turns to the original. For this
• reason, I shall make my account of Book V somewhat full,
with the object of indicating not only the whole content but
also the course of the proofs.
Of the Definitions the following are those which need
separate mention. The definition (3) of ratio as ' a sort of
relation (ttolol a-\eo-L?) in respect of size {tt^Xlkot^s) between
two magnitudes of the same kind' is as vague and of as
little practical use as that of a straight line ; it was probably
inserted for completeness' sake, and in order merely to aid the
conception of a ratio. Definition 4 (' Magnitudes are said to
have a ratio to one another which are capable, when multi-
plied, of exceeding one another ') is important not only because
1 Euclid, ed. Heib., vol. v, p. 282.
THE ELEMENTS. BOOK V 385
it shows that the magnitudes must be of the same kind,
but because, while it includes incommensurable as well as
commensurable magnitudes, it excludes the relation of a finite
magnitude to a magnitude of the same kind which is either
-infinitely great or infinitely small ; it is also practically equiva-
lent to the principle which underlies the method of exhaustion
now known as the Axiom of Archimedes. Most important
of all is the fundamental definition (5) of magnitudes which
are in the same ratio : ' Magnitudes are said to be in the same
ratio, the first to the second and the third to the fourth, when,
if any equimultiples whatever be taken of the first and third,
and any equimultiples whatever of the second and fourth, the
former equimultiples alike exceed, are alike equal to, or alike
fall short of, the latter equimultiples taken in corresponding
order.' Perhaps the greatest tribute to this marvellous defini-
tion is its adoption by Weierstrass as a definition of equal
numbers. For a most attractive explanation of its exact
significance and its absolute sufficiency the reader should turn
to De Morgan's articles on Ratio and Proportion in the Penny
Cyclopaedia. 1 The definition (7) of greater ratio is an adden-
dum to Definition 5 : ' When, of the equimultiples, the multiple
of the first exceeds the multiple of the second, but the
multiple of the third does not exceed the multiple of the
fourth, then the first is said to have a greater ratio to
the second than the third has to the fourth ' ; this (possibly
for brevity's sake) states only one criterion, the other possible
criterion being that, while the multiple of the first is equal
to that of the second, the multiple of the third is less than
that of the fourth. A proportion may consist of three or
four terms (Def s. 8, 9, 1 0) ; ' corresponding ' or ' homologous '
terms are antecedents in relation to antecedents and conse-
quents in relation to consequents (11). Euclid proceeds to
define the various transformations of ratios. Alternation
(ei>a\\d£, alternando) means taking the alternate terms in
the proportion a : b = c : d, i.e. transforming it into a:c = b:d
(12). Inversion (avdiraXiv, inversely) means turning the ratio
a:b into b:a (13). Composition of a ratio, arvvOeais Xoyov
(componendo is in Greek avvOevTL, 'to one who has compounded
1 Vol. xix (1841). I have largely reproduced the articles in The
Thirteen Books ofEucliaVs Elements, vol. ii, pp. 116-24.
1523 C C
386 EUCLID
or added', i.e. if one compounds or adds) is the turning of
a:b into (a + b):b (14). Separation, Siaipeo-is (SteXovTL =
separando) turns a: b into (a — b):b (15). Conversion, ava-
<TTpo<prj (dvao-TpeyjravTi = convertendo) turns a:b into a:a — b
(16). Lastly, e# aequali (sc. distantia), Si i(rov, and e# aequali
in disturbed proportion (eu rerapayfievr] avaXoyia) are defined
(17, 18). If a:b = A:B, b:c = B :C . . . k:l = K : L, then
the inference ex aequali is that a :l — A : L (proved in V. 22).
If again a : b = B : C and b:c = A:B, the inference e# aequali
in disturbed proportion is a : c = Jl : C (proved in V. 23).
In reproducing the content of the Book I shall express
magnitudes in general (which Euclid represents by straight
lines) by the letters a, b, c ... and I shall use the letters
m, n, p ... to express integral numbers : thus ma, mb are
equimultiples of a, b.
The first six propositions are simple theorems in concrete
arithmetic, and they are practically all proved by separating
into their units the multiples used.
' 1 . ma + mb + mc + . . . = m (a + b + c + . . .)•
5. ma — mb = m(a — b).
5 is proved by means of 1. As a matter of fact, Euclid
assumes the construction of a straight line equal to 1 /mth of
ma — mb. This is an anticipation of VI. 9, but can be avoided ;
for we can draw a straight line equal to m (a — b); then,
by 1 , m (a — b) + mb = ma, or ma—mb = m (a — b).
2. ma + na+pa+ ... = (m + n+p+ ...)a.
6. ma — na = (m — n) a.
Euclid actually expresses 2 and 6 by saying that ma±na is
the same multiple of a that mb±nb is of 6. By separation
of m, n into units he in fact shows (in 2) that
ma + na = (m -f n) a, and mb + nb = (m -f n) b.
6 is proved by means of 2, as 5 by means of 1.
3. If m .na, m.nb are equimultiples of na, nb, which are
themselves equimultiples of a, b, then m . na, m . nb are also
equimultiples of a, b.
By separating m } n into their units Euclid practically proves
that m . na = mn . a and m.nb = mn . b.
THE ELEMENTS. BOOK V 387
4. If a:b = c.d, then ma : nb = one : nd.
Take any equimultiples p. ma, p. vie of ona, mc, and any
equimultiples q . nb, q .nd of wfr, nd. Then, by 3, these equi-
multiples are also equimultiples of a, c and b, d respectively,
so that by Def. 5, since a : b = c : d,
p . ma > = < q.nb according as p . mc > = < q . nd,
whence, again by Def. 5, since p, q are any integers,
ma : nb = mc : nd.
7, 9. If a — b, then a:c = b:c'
, and conversely,
and c.a = c.b]
8, 10. If a > b, then a:c > b:c)
[ ; and conversely,
and c.b > c.a)
7 is proved by means of Def. 5. Take ma, mb equi-
multiples of a, b, and nc a multiple of c. Then, since a = b,
ma > — < nc according as mb > = < nc,
and nc > — < ma according as nc > = < mb,
whence the results follow.
8 is divided into two cases according to which of the two
magnitudes a — b, b is the less. Take m such that
m(a—b) >c or mb > c
in the two cases respectively. Next let nc be the first
multiple of c which is greater than mb or m(a — b) respec-
tively, so that
mb •
nc > . 7 > (n— l)c.
or m (a — b) ~~
Then, (i) since m (a — b) > c, we have, by addition, ma > nc.
(ii) since mb > c, we have similarly ma > nc.
In either case mb < nc, since in case (ii) m (a — b) > mb.
Thus in either case, by the definition (7) of greater ratio,
a:c > b:c,
and c : b > c:a.
The converses 9, 10 are proved from 7, 8 by reductio ad
absurdum.
c c 2
388 EUCLID
11. If a : b = c : d,
and c.cl = e:f,
then a:b = e:f.
Proved by taking any equimultiples of a, c, e and any other
equimultiples of b, d, f, and using Def . 5.
12. If a:b — c:d — e:f = ...
then a:b — (<z + c + e+ ...): (b + d+f+ ...).
Proved by means of V. 1 a,nd Def. 5, after taking equi-
multiples of a, c, e ... and other equimultiples of b, d,f....
13. If a:b = c:d,
and c:d > e:f,
then a : b > e :/.
Equimultiples mc, me of c, e are taken and equimultiples
nd, nf of d, f such that, while mc > nd, me is not greater
than nf (Def. 7). Then the same equimultiples ma, mc of
a, c and the same equimultiples nb, nd of 6, d are taken, and
Defs. 5 and 7 are used in succession.
14. If a:b = c:d, then, according as a > = < c, b > = < d.
The first case only is proved ; the others are dismissed with
1 Similarly '.
If a > c, amb > c:b. (8)
But a : b = c : d, whence (13) c:d > c:b, and therefore (10)
6 > d. ' •
15. a:b = ma:mb.
Dividing the multiples into their units, we have-m equal
ratios a:b\ the result follows by 1 2.
Propositions 16-19 prove certain cases of the transformation
of proportions in the sense of Defs. 12-16. The case of
inverting the ratios is omitted, probably as being obvious.
For, if a : b = c : d, the application of Def. 5 proves simul-
taneously that b : a = d : c.
*
16. If a:b — c:d,
then, alternando, a:c = bid.
Since a:b = ma : mb, and c:d = rtc: nd, ( 1 5)
THE ELEMENTS. BOOK V 389
we have ma : mb — nc : nd, (11)
whence (14), according as ma > = < nc, mb > = < nd;
therefore (Def . 5) a\c — b:d.
1 7. If a:b = c:d,
then, separando, (a — b):b = (c — d):d.
Take ma, mb, me, md equimultiples of all four magnitudes,
and nb, nd other equimultiples of b, d. It follows (2) that
(m + n) b, (m + n) d are also equimultiples of b, d.
Therefore, since a:b = c:d y
ma > = < (m + n)b according as mc > = < (m + n) d. (Del. 5)
Subtracting mb from both sides of the former relation and
md from both sides of the latter, we have (5)
m (a — b) > = < nb according as m(c — d) > = < nd.
Therefore (Def. 5) a — b:b = c — d:d.
(I have here abbreviated Euclid a little, without altering the
substance.)
18. If a : b = c : d,
then, componendo, (a + b):b = {c + d):d.
Proved by reductio ad absurdum. Euclid assumes that
a + b:b = (c + d): (d±x), if that is possible. (This implies
that to any three given magnitudes, two of which at least
are of the same kind, there exists a fourth proportional, an
assumption which is not strictly legitimate until the fact has
been proved by construction.)
Therefore, separando (17), a:b = (c + x) :(d±x),
whence (11), (c + x) : (d + x) = c : d, which relations are im-
possible, by 14.
19. If a:b = c:d,
then (a - c) : (b — d) = a: b.
Alternately (16),
a:c ■= b\d, whence (a — c) :c — (b— d) :d (17).
Alternately again, (a — c) : (b — d) = c:d (16);
whence (11) (a — c) : (b — d) = a : b.
390 EUCLID
The transformation convertendo is only given in an inter-
polated Porism to 19. But it is easily obtained by using 17
(separando) combined with alter nando (16). Euclid himself
proves it in X. 14 by using successively separando (17), inver-
sion and ex aequali (22).
The composition of ratios ex aequali and ex aequali in
disturbed proportion is dealt with in 22, 23, each of which
depends on a preliminary proposition.
20. If a:b = d:e,
and b:c = e:f,
then, ex aequali, according as a > = < c, d > = < f.
For, according as a > = < c, a : b > — < c : b (7, 8),
and therefore, by means of the above relations and 13, 11,
die > = < f : e,
and therefore again (9, 10)
d > = < /.
21. If a:b = e:f,
and b\c = d : e,
then, ex aequali in disturbed proportion,
according as a > — < c, d > = < /.
For, according as a > = < c, a : b > = < c : b (7, 8),
or e:f> = < e:d (13, 11),
and therefore d > = < / (9, 10).
22. If a:b =d:e,
and 6 : c = e :/,
then, ## aequali, a:c = d:f.
Take equimultiples wia, rac£ ; r<i>, ne ; jpc, p/, and it follows
that ma : nb = md : lie,) . .
and | 7i6 :pc = ne:pf )
There Tore (20), according as ma > — < pc, md > — < pf,
whence ' )ef . 5) a :c — d :/.
THE ELEMENTS. BOOK V 391
23. If a:b = e:/,
and b:c = d:e,
then, ea? aequali in disturbed proportion, a:c = d:f.
Equimultiples ma, mb, md and nc, ne, nf are taken, and
it is proved, by means of 11, 15, 16, that
ma : mb = ne : nf,
and mb : nc = md : ne,
whence (21) ma > = < nc according as md > — < nf,
and (Def. 5) a:c = d:f.
24. If a : c = d :f,
and also b : c = e :/,
then (a + b) : c = (d + e) :/.
Invert the second proportion to c:b = f:e, and compound
the first proportion with this (22) ;
therefore a:b = d:e.
Componendo, (a + b):b — (d+e) : e, which compounded (22)
with the second proportion gives (a + b) : c = (d + e) :/.
25. If a:b = c:d, and of the four terms a is the greatest
(so that d is also the least), a + d > b-{-c.
Since a:h = c(d,
a-c\b-d — a\b\ (19)
and, since a > b, (a — c)> (b — d). (16, 14)
Add c + d to each ;
therefore «+c£ > & + c.
Such slight defects as are found in the text of this great
Book as it has reached us, like other slight imperfections of
form in the Elements, point to the probability that the work
never received its final touches from Euclid's hand ; but they
can all be corrected without much difficulty, as Simson showed
in his excellent edition.
Book VI contains the application to plane geometry of the
general theory of proportion established in Book V. It begins
with definitions of ' similar rectilineal figures ' and of what is
392 EUCLID
meant by cutting a straight line ' in extreme and mean ratio \
The first and last propositions are analogous ; 1 proves that
triangles and parallelograms of the same height are to one
another as their bases, and 33 that in equal circles angles
, at the centre or circumference are as the arcs on which they
stand ; both use the method of equimultiples and apply
V, Def. 5 as the test of proportion. Equally fundamental
are 2 (that two sides of a triangle cut by any parallel to
the third side are divided proportionally, and the converse),
and 3 (that the internal bisector of an angle of a triangle cuts
the opposite side into parts which have the same ratio as the
sides containing the angle, and the converse) ; 2 depends
directly on 1 and 3 on 2. Then come the alternative con-
ditions for the similarity of two triangles : equality of all the
angles respectively (4), proportionality of pairs of sides in
order (5), equality of one angle in each with proportionality
of sides containing the equal angles (6), and the 'ambiguous
case ' (7), in which one angle is equal to one angle and the
sides about other angles are proportional. After the important
proposition (8) that the perpendicular from the right angle
in a right-angled triangle to the opposite side divides the
triangle into two triangles similar to the original triangle and
to one another, we pass to the proportional division of
straight lines (9, 10) and the problems of finding a third
proportional to two straight lines (11), a fourth proportional
to three (12), and a mean proportional to two straight lines
(13, the Book VI version of II. 14). In 14, 15 Euclid proves
the reciprocal proportionality of the sides about the equal
angles in parallelograms or triangles of equal area which have
1 one angle equal to one angle and the converse ; by placing the
equal angles vertically opposite to one another so that the sides
about them lie along two straight lines, and completing the
figure, Euclid is able to apply VI. 1. From 14 are directly
deduced 16, 17 (that, if four or three straight lines be propor-
tionals, the rectangle contained by the extremes is equal to
the rectangle contained by the two means or the square on the
one mean, and the converse). 18-22 deal with similar recti-
lineal figures ; 19 (with Porism) and 20 are specially important,
proving that similar triangles, and similar polygons generally,
are to one another in the duplicate ratio of corresponding
THE ELEMENTS. BOOK VI 393
sides, and that, if three straight lines are proportional, then,
as the first is to the third, so is the figure described on the first
to the similar figure similarly described on the second. The
fundamental case of the two similar triangles is prettily proved
thus. The triangles being ABC, DEF, in which B, E are equal
angles and BC, EF corresponding sides, find a third propor-
tional to BC, EF and measure it off along BC as BG ; join AG.
Then the triangles ABG, DEF have their sides about the equal
angles B, E reciprocally proportional and are therefore equal
(VI. 15); the rest follows from VI. 1 and the definition of
duplicate ratio (V, Def. 9).
Proposition 23 (equiangular parallelograms have to one
another the ratio compounded of the ratios of their sides) is
important in itself, and also because it introduces us to the
practical- use of the method of compounding, i.e. multiplying,
ratios which is of such extraordinarily wide application in
Greek geometry. Euclid has never defined ' compound ratio '
or the ' compounding ' of ratios ; but the meaning of the terms
B
K
L
M -
and the way to compound ratios are made clear in this pro-
position. The equiangular parallelograms are placed so that
two equal angles as BCD, GCE are vertically opposite at C.
Complete the parallelogram DCGH. Take any straight line K,
and (12) find another, L, such that
BC:CG = K:L,
and again another straight line M, such that
DC:CE=L:M.
Now the ratio compounded of K : L and X : if is K : M; there-
fore K : M is the ' ratio compounded of the ratios of the sides '.
And (ABCD) : (DCGH) = BC : CG, (1)
= K:L;
(DCGH) : (CEFG) = DC : CE ( 1 )
= L:M,
394
EUCLID
Therefore, ex aequali (V. 22),
(ABCD) : (GEFG) = K:M.
The important Proposition 25 (to construct a rectilineal figure
similar to one, and equal to another, given rectilineal figure) is
one of the famous problems alternatively associated with the
story of Pythagoras's sacrifice x ; it is doubtless Pythagorean.
The given figure (P, say) to which the required figure is to be
similar is transformed (I. 44) into a parallelogram on the same
base BC. Then the other figure (Q, say) to which the required
figure is to be equal is (I. 45) transformed into a parallelo-
gram on the base GF (in a straight line with BC) and of equal
height with the other parallelogram. Then (P) : (Q) = BC:CF
(1). It is then only necessary to take a straight line GH
a mean proportional between BC and GF, and to describe on
GH as base a rectilineal figure similar to P which has BC as
base (VI. 18). The proof of the correctness of the construction
follows from VI. 19 Por.
In 27, 28, 29 we reach the final problems in the Pythagorean
application of areas, which are the geometrical equivalent of
the algebraical solution of the most general form of quadratic
equation where that equation has a real and positive root.
Detailed notice of these propositions is necessary because of
their exceptional historic importance, which arises from the
fact that the method of these propositions was constantly used
by the Greeks in the solution of problems. They constitute,
for example, the foundation of Book X of the Elements and of
1 Plutarch, Non posse suaviter vivi secundum Epicurum, c. 11.
THE ELEMENTS. BOOK VI 395
the whole treatment of conic sections by Apollonius. The
problems themselves are enunciated in 28, 29 : ' To a given
straight line to apply a parallelogram equal to a given recti-
lineal figure and deficient (or exceeding) by a parallelogrammic
figure similar to a given parallelogram ' ; and 27 supplies the
Siopiauos, or determination of the condition of possibility,
which is necessary in the case of deficiency (28) : ' The given
rectilineal figure must (in that case) not be greater than the
parallelogram described on the half of the straight line and
similar to the defect.' We will take the problem of 28 for
examination.
We are already familiar with the notion of applying a
parallelogram to a straight line AB so that it falls short or
exceeds by a certain other parallelogram. Suppose that D is
the given parallelogram to which the defect in this case has to
be similar. Bisect AB in E, and on the half EB describe the
parallelogram GEBF similar and similarly situated to D.
Draw the diagonal GB and complete the parallelogram
HABF. Now, if we draw through any point T on HA a
straight line TR parallel to A B meeting the diagonal GB in
Q, and then draw PQS parallel to TA, the parallelogram TASQ
is a parallelogram applied to AB but falling short by a
parallelogram similar and similarly situated to B, since the
deficient parallelogram is QSBB which is similar to EF (24).
(In the same way, if T had been on HA produced and TR had
met GB produced in R, we should have had a parallelogram
applied to A B but exceeding by a parallelogram similar and
similarly situated to I).)
Now consider the parallelogram AQ falling short by SR
similar and similarly situated to I). Since (AO) = (ER), and
(OS) — (QF) y it follows that the parallelogram AQ is equal to
the gnomon UWV, and the problem is therefore that of
constructing the gnomon UWV such that its area is equal to
that of the given rectilineal figure C. The gnomon obviously
cannot be greater than the parallelogram EF, and hence the
given rectilineal figure C must not be greater than that
parallelogram. This is the 8iopio-ji6$ proved in 27.
Since the gnomon is equal to C, it follows that the parallelo-
gram GOQP which with it makes up the parallelogram EF is
equal to the difference between (EF) and C. Therefore, in
396 EUCLID
order to construct the required gnomon, we have only to draw
in the angle FGE the parallelogram GOQP equal to (EF) — C
and similar and similarly situated to D. This is what Euclid
in fact does ; he constructs the parallelogram LKNM equal to
(EF) — and similar and similarly situated to D (by means of
25), and then draws GOQP equal to it. The problem is thus
solved, TASQ being the required parallelogram.
To show the correspondence to the solution of a quadratic
equation, let AB — a, QS — x. and let b:c be the ratio of the
sides of B ; therefore SB — - x. Then, if m is a certain con-
c
stant (in fact the sine of an angle of one of the parallelograms),
(AQ) — m (ax — -x 2 ), so that the equation solved is
m (ax — - x 2 J = G.
The algebraical solution is x = t • ~ ± / st (t • -r — — )| •
Euclid gives only one solution (that corresponding to the
negative sign), but he was of course aware that there are two,
and how he could exhibit the second in the figure.
For a real solution we must have C not greater than
c a 2
m j- > —, which is the area of EF. This corresponds to Pro-
position 27.
We observe that what Euclid in fact does is to find the
parallelogram GOQP which is of given shape (namely such
that its area m . GO ,OQ = m. GO 2 -) and is equal to (EF) — C ;
c
c /c a G \
that is, he finds GO such that GO 2 = T ( y • — )• In other
o\b 4 m/
words, he finds the straight line equal to /]j\J'~7 )[ >
c a
and x is thus known, since x — GE — GO = -7 • GO.
Euclid's procedure, therefore, corresponds closely to the alge-
braic solution.
The solution of 29 is exactly similar, mutatis mutandis.
A solution is always possible, so that no Siopta-pos is required.
THE ELEMENTS. BOOKS VI VII 397
VI. 31 gives the extension of the Pythagorean proposition
I. 47 showing that for squares in the latter proposition we
may substitute similar plane figures of any shape whatever.
30 uses 29 to divide a straight line in extreme and mean
ratio (the same problem as II. 11).
Except in the respect that it is based on the new theory of
proportion, Book VI does not appear to contain any matter
that was not known before Euclid's time. Nor is the generali-
zation of I. 47 in VI. 31, for which Proclus professes such
admiration, original on Euclid's part, for, as we have already
seen (p. 191), Hippocrates of Chios assumes its truth for semi-
circles described on the three sides of a right-angled triangle.
We pass to the arithmetical Books, VII, VIII, IX. Book VII
begins with a set of definitions applicable in all the three
Books. They include definitions of a unit, a number, and the
following varieties of numbers, even, odd, even-times-even, even-
times-odd, odd-times-odd, prime, prime to one another, com-
posite, composite to one another, plane, solid, square, cube,
similar plane and solid numbers, and a perfect number,
definitions of terms applicable in the numerical theory of pro-
portion, namely a part (= a submultiple or aliquot part),
parts (= a proper fraction), multiply, and finally the defini-
tion of (four) proportional numbers, which states that ' num-
bers are proportional when the first is the same multiple, the
same part, or the same parts, of the second that the third is of
the fourth ', i.e. numbers a, b, c, d are proportional if, when
a = — b, c = — d, where m, n are any integers (although the
definition does not in terms cover the case where m > n).
The propositions of Book VII fall into four main groups.
1-3 give the method of finding the greatest common mea-
sure of two or three unequal numbers in essentially the same
form in which it appears in our text-books, Proposition 1
giving the test for two numbers being prime to one another,
namely that no remainder measures the preceding quotient
till 1 is reached. The second group, 4-19, sets out the
numerical theory of proportion. 4-10 are preliminary, deal-
ing with numbers which are ' a part ' or ' parts ' of other num-
bers, and numbers which are the same ' part ' or ' parts ' of
other numbers, just as the preliminary propositions of Book V
398 EUCLID
♦
deal with multiples and equimultiples. 11-14 are transforma-
tions of proportions corresponding to similar transformations
(separando, alternately, &c.) in Book V. The following are
the results, expressed with the aid of letters which here repre-
sent integral numbers exclusively.
If a : b = c : d (a > c. b > d), then
(a~c):(b-d) = a\b. (11)
If a: a' = b\U = c.c' . . ., then each of the ratios is equal to
(a + b + c+'...):(a' + b'+c' +...). (12)
If a:b = c:d, then a:c = b:d. (13)
If a:b = d:e and b : c = e:f, then, ex aequall,
a:c = d:f. (14)
If 1 : m = a : ma (expressed by saying that the third
number measures the fourth the same number of times that
the unit measures the second), then alternately
1 :a = m:ma. (15)
The last result is used to prove that ab = ba\ in other
words, that the order of multiplication is indifferent (16), and
this is followed by the propositions that b:c = ab:ac (17)
and that a:b = ac:bc (18), which are again used to prove
the important proposition (19) that, if a:b = c:d, then
ad = be, a theorem which corresponds to VI. 16 for straight
lines.
Zeuthen observes that, while it was necessary to use the
numerical definition of proportion to carry the numerical
theory up to this point, Proposition 1 9 establishes the necessary
point of contact between the two theories, since it is now
shown that the definition of proportion in V, Def. 5, has,
when applied to numbers, the same import as that in VII,
Def. 20, and we can henceforth without hesitation borrow any
of the propositions established in Book V. 1
Propositions 20, 21 about 'the least numbers of those which
have the same ratio with them ' prove that, if m, n are such
numbers and a, b any other numbers in the same ratio, m
1 Zeuthen, ' Sur la constitution des livres arithmetiques des Elements
d'Euclide ' ( Oversigt over det kgl. Danske Videnskabernes Selskabs Forhand-
linger, 1910, pp. 412, 413).
THE ELEMENTS. BOOKS VII-VIII 399
measures a the same number of times that it measures b, and
that numbers prime to one another are the least of those which
have the same ratio with them. These propositions lead up to
Propositions 22-32 about numbers prime to one another, prime
numbers, and composite numbers. This group includes funda-
mental theorems such as the following. If two numbers be
prime to any number, their product will be prime to the same
(24). If two numbers be prime to one another, so will their
squares, their cubes, and so on generally (27). If two numbers
be prime to one another, their sum will be prime to each
of them; and, if the sum be prime to either, the original
numbers will be prime to one another (28). Any prime number
is prime to any number which it does not measure (29). If two
numbers are multiplied, and any prime number measures the
product, it will measure one of the original numbers (30).
Any composite number is measured by some prime number
(31). Any number either is prime or is measured by some
prime number (32).
Propositions 33 to the end (39) are directed to the problem
of finding the least common multiple of two or three numbers ;
33 is preliminary, using the G. C. M. for the purpose of solving
the problem, ' Given as many numbers as we please, to find the
least of those which have the same ratio with them.'
It seems clear that in Book VII Euclid was following
earlier models, while no doubt making improvements in the
exposition. This is, as we have seen (pp. 215-16), partly con-
firmed by the fact that in the proof by Archytas of the
proposition that 'no number can be a mean between two
consecutive numbers ' propositions are presupposed correspond-
ing to VII. 20, 22, 33.
Book VIII deals largely with series of numbers ' in con-
tinued proportion ', i. e. in geometrical progression (Propositions
1-3, 6-7, 13). If the series in G. P. be
a n , a n ~ l b, a n ' 2 b 2 ,... a 2 b n - 2 , ab n ~\ b n ,
Propositions 1-3 deal with the case where the terms are the
smallest that are in the ratio a : b, in which case a n , b n are
prime to one another. 6-7 prove that, if a n does not measure
a n ~ Y b, no term measures any other, but if a n measures b n ,
it measures a n ~ l b. Connected with these are Propositions 14-17
400 EUCLID
proving that, according as a 2 does or does not measure b 2 ,
a does or does not measure b and vice versa; and similarly,
according as a 3 does' or does not measure b 3 , a does or does not
measure b and vice versa. 1 3 proves that, if a, b, c . . . are in
G. P., so are a 2 , b 2 , c 2 ... and a 3 , b 3 , c 3 ... respectively.
Proposition 4 is the problem, Given as many ratios as we
please, a:b, c:d... to find a series p, q, r, ... in the least
possible terms such that p:q = a:b, q:r = c:d, .... This is
done by finding the L. C. M., first of b, c, and then of other
pairs of numbers as required. The proposition gives the
means of compounding two or more ratios between numbers
in the same way that ratios between pairs of straight lines
are compounded in VI. 23 ; the corresponding proposition to
VI. 23 then follows (5), namely, that plane numbers have
to one another the ratio compounded of the ratios of their
sides.
Propositions 8-10 deal with the interpolation of geometric
means between numbers. If a:b = e :/, and there are n
geometric means between a and b, there are n geometric
means between e and/ also (8). If a n , a n_1 b ... ab n ~^, b n is a
G. P. of n + 1 terms, so that there are (n — 1 ) means between
a n , b n , there are the same number of geometric means between
1 and a n and between 1 and b 11 respectively (9); and con-
versely, if 1, a, a 2 ... a n and 1, b, b 2 ... b n are terms in G. P.,
there are the same number (n — 1) of means between a n , b n (10).
In particular, there is one mean proportional number between
square numbers (11) and between similar plane numbers (18),
and conversely, if there is one mean between two numbers, the
numbers are similar plane numbers (20) ; there are two means
between cube numbers (12) and between similar solid numbers
(19), and conversely, if there are two means between two num-
bers, the numbers are similar solid number^ (21). So far as
squares and cubes are concerned, these propositions are stated by
Plato in the Timaeus, and Nicomachus, doubtless for this reason,
calls them • Platonic '. Connected with them are the proposi-
tions that similar plane numbers have the same ratio as a square
has to a square (26), and similar solid numbers have the same
ratio as a cube has to a cube (27). A few other subsidiary
propositions need no particular mention.
Book IX begins with seven simple propositions such as that
THE ELEMENTS. BOOK IX 401
the product of two similar plane numbers is a square (1) and,
if the product of two numbers is a square number, the num-
bers are similar plane numbers (2) ; if a cube multiplies itself
or another cube, the product is a cube (3, 4); if a 3 B is a
cube, B is a cube (5) ; if A 2, is a cube, A is a cube (6). Then
follow six propositions (8-13) about a series of terms in geo-
metrical progression beginning with 1 . If 1 , a, b, c . . . k are
n terms in geometrical progression, then (9), if a is a square
(or a cube), all the other terms b, c, ... k are squares (or
cubes) ; if a is not a square, then the only squares in the series
are the term after a, i. e. b, and all alternate terms after b ; if
a is not a cube, the only cubes in the series are the fourth
term (c), the seventh, tenth, &c, terms, being terms separated
by two throughout ; the seventh, thirteenth, &c, terms (leaving
out five in each case) will be both square and cube (8, 10).
These propositions are followed by the interesting theorem
that, if 1, a Xi a 2 ... a n ... are terms in geometrical progression,
and if a r , a n are any two terms where r<n, a r measures a w
and a n — a r .a n _ r (11 and Por.) ; this is, of course, equivalent
to the formula a m+n = a m . a n . Next it is proved that, if the
last term & in a series 1, a, b, c ... k in geometrical progression
is measured by any primes, a is measured by the same (12) ;
and, if a is prime, k will not be measured by any numbers
except those which have a place in the series (13). Proposi-
tion 14 is the equivalent of the important theorem that a
number can only be resolved into prime factors in one way.
Propositions follow to the effect that, if a, b be prime to one
another, there can be no integral third proportional to them
(16) and, if a,b,c ...k be in G. P. and a, k are prime to one
another, then there is no integral fourth proportional to a, b, k
(17). The conditions for the possibility of an integral third
proportional to two numbers and of an integral fourth propor-
tional to three are then investigated (18, 19)". Proposition 20
is the important proposition that the number of prime numbers
is infinite, and the proof is the same as that usually given in
our algebraical text-books. After a number of easy proposi-
tions about odd, even, ' even-times-odd ', ' even-times-even '
numbers respectively (Propositions 21-34), we have two im-
portant propositions which conclude the Book. Proposition 35
gives the summation of a G. P. of n terms, and a very elegant
1823 D d
402 EUCLID
solution it is. Suppose that a v a 2 , a. 6i ... a n+1 are n+ 1 terms
in G. P. ; Euclid proceeds thus :
We have
and, separando,
a n+l _ a n __ *h
a n a n-i a i
j
(l n + l Ct> n Ct n tt w _j _ CL 2 a \
Ct n a n-i a i
Adding antecedents and consequents, we have (VII. 12)
<% n+ i ttj a 2 — a x
^n ' ^n — 1 ' ••• '^i ^i
which gives a n + a n _ x + ... + a x or S n .
The last proposition (36) gives the criterion for perfect
numbers, namely that, if the sum of any number of terms of
the series 1, 2, 2 2 ... 2 n is prime, the product of the said sum
and of the last term, viz. (1+ 2 + 2 2 + ... + 2 n ) 2 n , is a perfect
number, i. e. is equal to the sum of all its factors.
It should be added, as regards all the arithmetical Books,
that all numbers are represented in the diagrams as simple
straight lines, whether they are linear, plane, solid, or any
other kinds of numbers ; thus a product of two or more factors
is represented as a new straight line, not as a rectangle or a
solid.
Book X is perhaps the most remarkable, as it is the most
perfect in form, of all the Books of the Elements. It deals
with irrationals, that is to say, irrational straight lines in rela-
tion to any particular straight line assumed as rational, and
it investigates every possible variety of straight lines which
can be represented by */{*/a+ Vb), where a, b are two com-
mensurable lines. The theory was, of course, not invented by
Euclid himself. On the contrary, we know that not only the
fundamental proposition X. 9 (in which it is proved that
squares which have not to one another the ratio of a square
number to a square number have their sides incommen-
surable in length, and conversely), but also a large part of
the further development of the subject, was due to Theae-
tetus. Our authorities for this are a scholium to X. 9 and a
passage from Pappus's commentary on Book X preserved
in the Arabic (see pp. 154-5, 209-10, above). The passage
THE ELEMENTS. BOOKS IX-X 403
of Pappus goes on to speak of the share of Euclid in the
investigation :
' As for Euclid, he set himself to give rigorous rules, which he
established, relative to commensurability and incommensura-
bility in general ; he made precise the definitions and the
distinctions between rational and irrational magnitudes, he set
out a great number of orders of irrational magnitudes, and
finally he made clear their whole extent.'
As usual, Euclid begins with definitions. ' Commensurable '
magnitudes can be measured by one and the same measure ;
' incommensurable ' magnitudes cannot have any common
measure (l). Straight lines are 'commensurable in square'
when the squares on them can be measured by the same area,
but ' incommensurable in square ' when the squares on them
have no common measure (2). Given an assigned straight
line, which we agree to call ' rational ', any straight line which
is commensurable with it either in length or in square only is
also called rational ; but any straight line which is incommen-
surable with it (i.e. not commensurable with it either in length
or in square) is ' irrational ' (3). The square on the assigned
straight line is ' rational ', and any area commensurable with
it is ' rational ', but any area incommensurable with it is
' irrational ', as also is the side of the square equal to that
area (4). As regards straight lines, then, Euclid here takes
a wider view of 'rational' than we have met before. If a
straight line p is assumed as rational, not only is — p also
' rational ' where m, n are integers and m/n in its lowest terms
is not square, but any straight line is rational which is either
commensurable in length or commensurable in square only
l r fli
with p; that is, / — •/> is rational according to Euclid. In
IT)
the case of squares, p 2 is of course rational, and so is— p 2 ; but
I'Yb
- • p 2 is not rational, and of course the side of the latter
Yb
•J
1 7YL
square */ — ' P * s irrational, as are all straight lines commen-
surable neither in length nor in square with p, e.g. Va± Vb
or (■/& + V\).p.
D d 2
404 EUCLID
The Book begins with the famous proposition, on which the
' method of exhaustion ' as used in Book XII depends, to the
effect that, if from any magnitude there be subtracted more
than its half (or its half simply), from the remainder more than
its half (or its half), and so on continually, there will at length
remain a magnitude less than any assigned magnitude of the
same kind. Proposition 2 uses the process for finding the
G. C. M. of two magnitudes as a test of their commensurability
or incommensurability: they are incommensurable if the process
never comes to an end, i.e. if no remainder ever measures the
preceding divisor ; and Propositions 3, 4 apply to commen-
surable magnitudes the method of finding the G. C. M. of two
or three numbers as employed in VII. 2, 3. Propositions 5
to 8 show that two magnitudes are commensurable or incom-
mensurable according as they have or have not to one another
the ratio of one number to another, and lead up to the funda-
mental proposition (9) of Theaetetus already quoted, namely
that the sides of squares are commensurable or incommen-
surable in length according as the squares have or have not to
one another the ratio of a square number to a square number,
and conversely. Propositions 11-16 are easy inferences as to
the commensurability or incommensurability of magnitudes
from the known relations of others connected with them ;
e. g. Proposition 1 4 proves that, if a : b = c : d, then, according
as \/(a 2 — b 2 ) is commensurable or incommensurable with a.
*/(c 2 —d 2 ) is commensurable or incommensurable with c.
Following on this, Propositions 17, 18 prove that the roots of
the quadratic equation ax — x 2 = 6 2 /4 are commensurable or
incommensurable with a according as V(a 2 — b 2 ) is commen-
surable or incommensurable with a. Propositions 19-21 deal
with rational and irrational rectangles, the former being
contained by straight lines commensurable in length, whereas
rectangles contained by straight lines commensurable in square
only are irrational. The side of a square equal to a rectangle
of the latter kind is called medial ; this is the first in Euclid's
classification of irrationals. As the sides of the rectangle may
be expressed as p, pVh, where p is a rational straight line,
the medial is k^p. Propositions 23-8 relate to medial straight
lines and rectangles ; two medial straight lines may be either
commensurable in length or commensurable in square only :
THE ELEMENTS. BOOK X 405
thus k*p and f A&*p are commensurable in length, while Jc^p
and */\ . k*p are commensurable in square only : the rectangles
formed by such pairs are in general medial, as \l<$p 2 and
V\ . k%p 2 ; but if \/A = k' */ k in the second case, the rectangle
(k'lcp 2 ) is rational (Propositions 24, 25). Proposition 26 proves
that the difference between two medial areas cannot be
rational ; as any two medial areas can be expressed in the
form Vk . p 2 , V\ . p 2 , this is equivalent to proving, as we do in
algebra, that ( Vk — VX) cannot be equal to k' '. Finally,
Propositions 27, 28 find medial straight lines commensurable
in square only (1) which contain a rational rectangle, viz. k*p,
k^p, and (2) which contain a medial rectangle, viz. k%p,\hp/k*. It
should be observed that, as p may take either of the forms a
or VA, a medial straight line may take the alternative forms
V(aVB) or V(AB), and the pairs of medial straight lines just
mentioned may take respectively the forms
and (2) V(aVB h J(£.) or V { AB), J(bg).
I shall henceforth omit reference to these obvious alternative
forms. Next follow two lemmas the object of which is to find
(1) two square numbers the sum of which is a square, Euclid's
solution being
„ o /mnp 2 — mnq\ 2 /mnp 2 + mnq 2 \ 2
map* . Tn/nq* + ( — ) = ( — ~ j ?
where mwp 2 , mnq 2 are either both odd or both even, and
(2) two square numbers the sum of which is not square,
Euclid's solution being
m r p 2 . mq 2 , { — 1 j •
Propositions 29-35 are problems the object of which is to find
(a) two rational straight lines commensurable in square only,
(b) two medial straight lines commensurable in square only,
(c) two straight lines incommensurable in square, such that
the difference or sum of their squares and the rectangle
406 EUCLID
contained by them respectively have certain characteristics.
The solutions are
(a) x, y rational and commensurable in square only.
Prop. 29: p, pV{\ — k 2 ) [V(x 2 —y 2 ) commensurable with as].
„ 30: p, p/ V(\ + k 2 ) [V(x 2 — y 2 ) incommensurable with as].
(b) x, y medial and commensurable in square only.
Prop. 31 : p (1 — k 2 )±, p(\—k 2 )% [xy rational, V(x 2 — y 2 ) commen-
surable with as] ;
p/(l+/c 2 )±, p/(l+k 2 )* [xy rational, V(x 2 — y 2 ) incom-
mensurable with x\.
„ 32: p\%, p\*V(l—k 2 ) [xy medial, V{x 2 — y 2 ) commensur-
able with x] ;
p\i, p\*/V(l +k 2 ) [xy medial, V(x 2 — y 2 ) incommen-
surable with as].
(c) x, y incommensurable in square.
Prop.33:-f o /(l + -iL=), -f o lU-^L^
F V2^\ Vl+k 2 ' V2\J\ Vl+W
[(x 2 +y 2 ) rational, xy medial].
34
P
_^./ {V (i +**) + *},
[x 2 -\-y 2 medial, xy rational].
p\* /j k ) p\* K k |
V2 <sj ( + V(l+k 2 )\' V2\]\ " V(l+k 2 )\
[x 2 + y 2 and xy both medial and
incommensurable with one another].
With Proposition 36 begins Euclid's exposition of the several
compound irrationals, twelve in number. Those which only
differ in the sign separating the two component parts can be
35
THE ELEMENTS. BOOK X 407
taken together. The twelve compound irrationals, with their
names, are as follows :
(AJ | Binomial, p+ y/k.p (Prop. 36)
(A 2 ) (Apotome, p <*> Vk.p (Prop. 73)
(B,) (First bimedial ) ,, ., ._
, D \ L- . t ! k*p + klp Props. 37, 74
(B 2 ) (First apotome of a medial J r ^ r r
(C,) (Second bimedial ) 7i X^p ,_ nn ■
v 17 J L A?*p + -f- (Props. 38, 75)
(C 2 ) (Second apotome of a medial] "* &*
(A) (Major | _p^ / / ft v ^p_ // _ __J\
(D 2 ) (Minor] V2 ^/ V -/l +#■/ ~ V2 /y/ V -/l + fc»/
(Props. 39, 76)
(EJ /Side of a rational plus\ p 7/ *- — t^ 7
! • -i — V (v 1+ft! + /£)
a medial area a/2(1+/c 2 )
(i? 2 ) < That which c produces ' Y p
with a rational area i / 9 /, , pE\
\ a medial whole J
with a rational area I ± / n/ , , ,. 2X ^(</l +& 2 -&)
(Props. 40, 77)
(^\) /Side of the sum of two\ pA* l / h
cle oi the sum or two\ pA 4 J / fc \
medial areas \ V2\f \ v 7 1 + k 2 '
7F+F 2 >
{F 2 ) ^That which 'produces'- p\l i, & \
with a medial area ± V2\l\" yfTfTp/
\ a medial whole j /r , „„.
(Props. 41, 78).
As regards the above twelve compound irrationals, it is
to be noted that
A lt A 2 are the positive roots of the equation
x i -2(l+k)p 2 .x 2 + (l-Lfp A = 0;
B Y , B 2 are the positive roots of the equation
x i -2Vk(l+k)p 2 .x 2 + k(l-k) 2 p* = 0;
C\ , C 2 are the positive roots of the equation
ar — 2 — Tj- p' 5 . as 2 H t— ^- p 4 == ;
v /v A;
408 EUCLID
D Y , D 2 are the positive roots of the equation
E x , E 2 are the positive roots of the equation
F lf F 2 are the positive roots of the equation
k*
« 4 -2^A.a;y + A - — r „p
r 1 + kr r
4 _
0.
Propositions 42-7 prove that each of the above straight lines,
made up of the stim of two terms, is divisible into its terms
in only one way. In particular, Proposition 42 proves the
equivalent of the well-known theorem in algebra that,
if a + Vb = x + Vy, then a = x, b = y;
and if Va + Vb = Vx + Vy f
then a = x, b — y (or a — y, b = x).
Propositions 79-84 prove corresponding facts in regard to
the corresponding irrationals with the negative sign between
the terms: in particular Proposition 79 shows that,
if a — Vb = x — Vy, then a — x, b = y;
and if v a— Vb = Vx— Vy, then a — x, b = y.
The next sections of the Book deal with binomials and
apotomes classified according to the relation of their terms to
another given rational straight line. There are six kinds,
which are first defined and then constructed, as follows :
(a.) ( First binomial ) _ _ .. , v _
K \ „. [ kp + kpV(l-\ 2 ): (Props. 48, 85)
(a 2 ) \ First apotome J H ~ H V ; ' V 1 ' '
(&) (Second binomial) kp
, a [ • o , • //i X 2i + k P : > (Props. 49, 86)
(P 2 ) (Second apotomej V(l-\')- r ' r
(y,) ( Third binomial ) . ,, ,
, \ 1m,. , \mVk.p+mVk.pV(\-\ 2 ))
y 2 ) 1 Third apotome J H ~ m '
y/2f K l ' (Props. 50, 87)
(Sj) [Fourth binomial
(8 2 ) (Fourth apotome
w
THE ELEMENTS. BOOK X
409
&/o±
a/(1+A)
; (Props. 51, 88)
Fifth binomial
Fifth apotome
kpS(l+\)±kp; (Props. 52, 89)
Sixth binomial )
Vk.p±V\.p. (Prop. 53, 90)
tfi) J
(&) I Sixth apotome ]
Here again it is to be observed that these binomials and
apotomes are the greater and lesser roots respectively of
certain quadratic equations,
a x , a 2 being the roots of x 2 — 2kp.x + \ 2 k 2 p 2 — 0,
ft, ft
33 i
Yv Y2
33 3
#i> ^2
33 3
*1> 6 2
33 33
&> &2
33 33
X* —
2k
x +
A 2
k 2 p 2 = 0,
V(l-A 2 ) " 1-A 2
x 2 — 2mVk.px + \ 2 m 2 kp' i = 0,
1 + A
x 2 -2kp\/(l+\).x + \k 2 p 2 = 0,
x 2 -2\/k.px + (k-\)p 2 = 0.
The next sets of propositions (54-65 and 91-102) prove the
connexion between the first set of irrationals (A 1} A 2 ... F li F 2 )
and the second set (a x , a 2 ... &, £ 2 ) respectively. It is shown
e.g., in Proposition 54, that the side of a square equal to the
rectangle contained by p and the first binomial oc ± is a binomial
of the type A lf and the same thing is proved in Proposition 91
for the first apotome. In fact
V{p(kp±k P VT^\ 2 )} =>-/{i*(i+x)j ±pV{ik(i-\)}.
Similarly A/(pft), V(pft) <^re irrationals of the type B lt B 2
respectively, and so on.
Conversely, the square on A 1 or A 2 , if applied as a rectangle
to a rational straight line (0-, say), has for its breadth a binomial
or apotome of the types oc x , ol 2 respectively (60, 97).
2
In fact (p ± Vk . pf/a- = C { (1 + k) ± 2 Vk} ,
and -ft 2 , B 2 2 are similarly related to irrationals of the type
ft, ft, and so on.
410 EUCLID
Propositions 66-70 and Propositions 103-7 prove that
straight lines commensurable in length with A x , A 2 ... F 1} b\,
respectively are irrationals of the same type and order.
Propositions 71, 72, 108-10 show that the irrationals
A x , A 2 ... F lt F 2 arise severally as the sides of squares equal
to the sum or difference of a rational and a medial area, or the
sum or difference of two medial areas incommensurable with
one another. Thus kp 2 + \/\ . p 2 is the sum or difference of a
rational and a medial area, Vk.p 2 + VX.p 2 is the sum or
difference of two medial areas incommensurable with one
another provided that Vk and VX are incommensurable, and
the propositions prove that
v {kp 2 + VX . p 2 ) and V( Vk .p 2 ±VX. p 2 )
take one or other of the forms A lf A 2 ... F lt F 2 according to
the different possible relations between k, X and the sign
separating the two terms, but no other forms.
Finally, it is proved at the end of Proposition 72, in Proposi-
tion 111 and the explanation following it that the thirteen
irrational straight lines, the medial and the twelve other
irrationals A ly A 2 ... F x , F 2 , are all different from one another.
E.g. (Proposition 111) a binomial straight line cannot also be
an apotome ; in other words, Vx + Vy cannot be equal to
Vx' — </y', and x + Vy cannot be equal to x / — */y'. We
prove the latter proposition by squaring, and Euclid's proce-
dure corresponds exactly to this. Propositions 112-14 prove
that, if a rectangle equal to the square on a rational straight
line be applied to a binomial, the other side containing it is an
apotome of the same order, with terms commensurable with
those of the binomial and in the same ratio, and vice versa ;
also that a binomial and apotome of the same order and with
terms commensurable respectively contain a rational rectangle.
Here we have the equivalent of rationalizing the denominators
c- c 2
of the fractions —^ Tf) or —. h by multiplying the
V A + V Jo a+Vn
numerator and denominator by VAT VB or a+ VB respec-
tively. Euclid in fact proves that
cr 2 /(p+ Vk. p) — Xp- Vk.Xp (k < 1),
and his method enables us to see that X = cr 2 / (p 2 — kp 2 ).
Proposition 115 proves that from a medial straight line an
THE ELEMENTS. BOOK X 411
infinite number of other irrational straight lines arise each
of which is different from the preceding, h^p being medial,
we take another rational straight line a and find the mean
proportional V(l$pa)) this is a new irrational. Take the
mean between this and </, and so on.
I have described the contents of Book X at length because
it is probably not well known to mathematicians, while it is
geometrically very remarkable and very finished. As regards
its object Zeuthen has a remark which, I think, must come
very near the truth. ' Since such roots of equations of the
second degree as are incommensurable with the given magni-
tudes cannot be expressed by means of the latter and of num-
bers, it is conceivable that the Greeks, in exact investigations,
introduced no approximate values, but worked on with the
magnitudes they had found, which were represented by
straight lines obtained by the construction corresponding to
the solution of the equation. That is exactly the same thing
which happens when we do not evaluate roots but content
ourselves with expressing them by radical signs and other
algebraical symbols. But, inasmuch as one straight line looks
like another, the Greeks did not get the same clear view of
what they denoted (i. e. by simple inspection) as our system
of symbols assures to us. For this reason then it was neces-
sary to undertake a classification of the irrational magnitudes
which had been arrived at by successive solutions of equations
of the second degree.' That is, Book X formed a repository
of results to which could be referred problems depending on
the solution of certain types of equations, quadratic and
biquadratic but reducible to quadratics, namely the equations
x 2 ± 2 px . p ± v . p 2 = 0,
and x* ±2 px 2 . p 2 ± v . /> 4 = 0,
where p is a rational straight line and p, v are coefficients.
According to the values of p, v in relation to one another and
their character (p, but not v, may contain a surd such as
\/m or \Z{m/n)) the two positive roots of the first equations are
the binomial and apotome respectively of some one of the
orders ' first ', ' second ',...' sixth ', while the two positive
roots of the latter equation are of some one of the other forms
of irrationals (A 1 , A 2 ), (B 1} B 2 ) ... (F lt F 2 ).
412 EUCLID
Euclid himself, in Book XIII, makes considerable use of the
second part of Book X dealing with apotomes ; he regards a
straight line as sufficiently defined in character if he can say
that it is, e.g., an apotome (XIII. 17), & first apotome (XIII. 6),
a minor straight line (XIII. 11). So does Pappus. 1
Our description of Books XI-XIII can be shorter. They
deal with geometry in three dimensions. The definitions,
belonging to all three Books, come at the beginning of Book XI.
They include those of a straight line, or a plane, at right angles
to a plane, the inclination of a plane to a plane (dihedral angle),
parallel planes, equal and similar solid figures, solid angle,
pyramid, prism, sphere, cone, cylinder and parts of them, cube,
octahedron, icosahedron and dodecahedron. Only the defini-
tion of the sphere needs special mention. Whereas it had
previously been defined as the figure which has all points of
its surface equidistant from its centre, Euclid, with an eye to
his use of it in Book XIII to ' comprehend ' the regular solids
in a sphere, defines it as the figure comprehended by the revo-
lution of a semicircle about its diameter.
The propositions of Book XI are in their order fairly
parallel to those of Books I and VI on plane geometry. First
we have propositions that a straight line is wholly in a plane
if a portion of it is in the plane (1), and that two intersecting
straight lines, and a triangle, are in one plane (2). Two
intersecting planes cut in a straight line (3). Straight lines
perpendicular to planes are next dealt with (4-6, 8, 11-14),
then parallel straight lines not all in the same plane (9, 10, 15),
parallel planes (14, 16), planes at right angles to one another
(18, 19), solid angles contained by three angles (20, 22, 23, 26)
or by more angles (21). The rest of the Book deals mainly
with parallelepipedal solids. It is only necessary to mention
the more important propositions. Parallelepipedal solids on the
same base or equal bases and between the same parallel planes
(i.e. having the same height) are equal (29-31). Parallele-
pipedal solids of the same height are to one another as their
bases (32). Similar parallelepipedal solids are in the tripli-
cate ratio of corresponding sides (33). In equal parallele-
pipedal solids the bases are reciprocally proportional to their
heights and conversely (34). If four straight lines be propor-
1 Cf. Pappus, iv, pp. 178, 182.
THE ELEMENTS. BOOKS XI-XII 413
tional, so are parallelepipedal solids similar and similarly
described upon them, and conversely (37). A few other
propositions are only inserted because they are required as
lemmas in later books, e.g. that, if a cube is bisected by two
planes each of which is parallel to a pair of opposite faces, the
common section of the two planes and the diameter of the
cube bisect one another (38).
The main feature of Book XII is the application of the
method of exhaustion, which is used to prove successively that
circles are to one another as the squares on their diameters
(Propositions 1, 2), that pyramids of the same height and with
triangular bases are to one another as the bases (3-5), that
any cone is, in content, one third part of the cylinder which
has the same base with it and equal height (10), that cones
and cylinders of the same height are to one another as their
bases (11), that similar cones and cylinders are to one another
in the triplicate ratio of the diameters of their bases (12), and
finally that spheres are to one another in the triplicate ratio
of their respective diameters (16-18). Propositions 1, 3-4 and
16-17 are of course preliminary to the main propositions 2, 5
and 18 respectively. Proposition 5 is extended to pyramids
with polygonal bases in Proposition 6. Proposition 7 proves
that any prism with triangular bases is divided into three
pyramids with triangular bases and equal in content, whence
any pyramid with triangular base (and therefore also any
pyramid with polygonal base) is equal to one third part of
the prism having the same base and equal height. The rest
of the Book consists of propositions about pyramids, cones,
and cylinders similar to those in Book XI about parallele-
pipeds and in Book VI about parallelograms : similar pyra-
mids with triangular bases, and therefore also similar pyramids
with polygonal bases, are in the triplicate ratio of correspond-
ing sides (8) ; in equal pyramids, cones and cylinders the bases
are reciprocally proportional to the heights, and conversely
(9, 15).
The method of exhaustion, as applied in Euclid, rests upon
X. 1 as lemma, and no doubt it will be desirable to insert here
an example of its use. An interesting case is that relating^to
the pyramid. Pyramids with triangular bases and of the same
height, says Euclid, are to one another as their bases (Prop. 5).
414 EUCLID
It is first proved (Proposition 3) that, given any pyramid, as
A BCD, on the base BCD, if we bisect the six edges at the
points E, F, G, H, K, L, and draw the straight lines shown in
the figure, we divide the pyramid A BCD into two equal
prisms and two equal pyramids AFGE, FBHK similar to the
original pyramid (the equality of the prisms is proved in
XI. 39), and that the sum of the two prisms is greater than
half the original pyramid. Proposition 4 proves that, if each
of two given pyramids of the same height be so divided, and
if the small pyramids in each are similarly divided, then the
smaller pyramids left over from that division are similarly
divided, and so on to any extent, the sums of all the pairs of
prisms in the two given pyramids respectively will be to one
another as the respective bases. Let the two pyramids and
their volumes be denoted by P, P' respectively, and their bases
by B, B' respectively. Then, if B : B' is not equal to P : P', it
must be equal to P : IF, where W is some volume either less or
greater than P' .
I. Suppose W < P'.
By X. 1 we can divide P' and the successive pyramids in
it into prisms and pyramids until the sum of the small
pyramids left over in it is less that P' — W, so that
P' > (prisms in P f ) > W.
Suppose this done, and P divided similarly.
Then (XII. 4)
(sum of prisms in P) : (sum of prisms in P') — B\ B f
= P : W, by hypothesis.
But P > (sum of prisms in P) :
therefore W > (sum of prisms in P').
THE ELEMENTS. BOOKS XII-XIII 415
But W is also less than the sum of the prisms in P / : which
is impossible.
Therefore W is not less than P'.
II. Suppose W > P'.
We have, inversely,
&:B= W:P
= P' :V, where V is some solid less than P.
But this can be proved impossible, exactly as in Part I.
Therefore W is neither greater nor less than P', so that
B:B / =P:P / .
We shall see, when we come to Archimedes, that he extended
this method of exhaustion. Instead of merely taking the one
approximation, from underneath as it were, by constructing
successive figures within the figure to be measured and so
exhausting it, he combines with this an approximation from
outside. He takes sets both of inscribed and circumscribed
figures, approaching from both sides the figure to be measured,
and, as it were, compresses them into one, so that they coincide
as nearly as we please with one another and with the curvi-
linear figure itself. The two parts of the proof are accordingly
separate in Archimedes, and the second is not merely a reduction
to the first.
The object of Book XIII is to construct, and to ' comprehend
in a sphere ', each of the five regular solids, the pyramid
(Prop. 13), the octahedron (Prop. 14), the cube (Prop. 15),
the icosahedron (Prop. 16) and the dodecahedron (Prop. 17);
' comprehending in a sphere ' means the construction of the
circumscribing sphere, which involves the determination of
the relation of a ' side ' (i. e. edge) of the solid to the radius
of the sphere ; in the case of the first three solids the* relation
is actually determined, while in the case of the icosahedron
the side of the figure is shown to be the irrational straight
line called ' minor ', and in the case of the dodecahedron an
' apotome \ The propositions at the beginning of the Book
are preliminary. Propositions 1-6 are theorems about straight
lines cut in extreme and mean ratio, Propositions 7, 8 relate
to pentagons, and Proposition 8 proves that, if, in a regular
pentagon, two diagonals (straight lines joining angular points
416 EUCLID
next but one to each other) are drawn intersecting at a point,
each of them is divided at the point in extreme and mean
ratio, the greater segment being equal to the side of the pen-
tagon. Propositions 9 and 1 relate to the sides of a pentagon,
a decagon and a hexagon all inscribed in the same circle,
and are preliminary to proving (in Prop. 1 1 ) that the side of
the inscribed pentagon is, in relation to the diameter of the
circle, regarded as rational, the irrational straight line called
: minor '. If p, d, h be the sides of the regular pentagon,
decagon, and hexagon inscribed in the same circle, Proposition 9
proves that h + d is cut in extreme and mean ratio, h being the
greater segment ; this is equivalent to saying that (r + d)d = r 2 ,
where r is the radius of the circle, or, in other words, that
d = -§r(\/5— 1). Proposition 10 proves that r p l = h 2 + d 2 or
r 2 + d 2 , whence we obtain p = ±rV(lO — 2 \/5). Expressed as
a ' minor ' irrational straight line, which Proposition 1 1 shows
it to be, p = |?V(5 + 2\/5)-ir \/(5 -2 a/5).
The constructions for the several solids, which have to be
inscribed in a given sphere, may be briefly indicated, thus :
1. The regular pyramid or tetrahedron.
Given D, the diameter of the sphere which is to circum-
scribe the tetrahedron, Euclid draws a circle with radius r
such that r 2 = ^D.^D, or r = -§\/2 . D, inscribes an equi-
lateral triangle in the circle, and then erects from the centre
of it a straight line perpendicular to its plane and of length
§D. The lines joining the extremity of the perpendicular to
the angular points of the equilateral triangle determine the
tetrahedron. Each of the upstanding edges (x, say) is such
that x 2 = r 2 + §D 2 = 3r 2 , and it has been proved (in XIII. 12)
that the square on the side of the triangle inscribed in the
circle is also 3r 2 . Therefore the edge a of the tetrahedron
=V3.r = §^6.D.
2. The octahedron.
If D be the diameter of the circumscribing sphere, a square
is inscribed in a circle of diameter D, and from its centre
straight lines are drawn in both directions perpendicular to
its plane and of length equal to the radius of the circle or half
the diagonal of the square. Each t)f the edges which stand up
from the square = V 2 . ^D, which is equal to the side of the
THE ELEMENTS. BOOK XIII 417
square. Each of the edges a of the octahedron is therefore
equal to \/2 . \T>.
3. The cube.
D being the diameter of the circumscribing sphere, draw
a square with side a such that a 2 = D . ^D, and describe a cube
on this square as base. The edge a = § 73 . D.
4. The icosahedron.
Given D, the diameter of the sphere, construct a circle with
radius r such that r 2 = D . ±D. Inscribe in it a regular
decagon. Draw from its angular points straight lines perpen-
dicular to the plane of the circle and equal in length to its
radius r; this determines the angular points of a regular
decagon inscribed in an equal parallel circle. By joining
alternate angular points of one of the decagons, describe a
regular pentagon in the circle circumscribing it, and then do
the same in the other circle but so that the angular points are
not opposite those of the other pentagon. Join the angular
points of one pentagon to the nearest angular points of the
other ; this gives ten triangles. Then, if p be the side of each
pentagon, d the side of each decagon, the upstanding sides
of the triangles (= x, say) are given by x 2 — d 2 + r 2 = p 2
(Prop. 1 0) ; therefore the ten triangles are equilateral. We
have lastly to find the common vertices of the five equilateral
triangles standing on the pentagons and completing the icosa-
hedron. If (7, C be the centres of the parallel circles, CC is
produced in both directions to X, Z respectively so that
CX — G'Z = d (the side of the decagon). Then again the
upstanding edges connecting to X, Z the angular points of the
two pentagons respectively ( — x, say) are given by
x 2 = v 2 + d 2 = p 2 .
Hence each of the edges
a = p = |?V(10-2/5) = 7 — r - 7(10-2 \/5)
u v o
= ?\D</ '{10(5-V5)}.
It is finally shown that the sphere described on XZ as
diameter circumscribes the icosahedron, and
kZ = r + 2d = r + r(V5-l) = r.V5=D.
1523 E e
418
EUCLID
5. The dodecahedron.
We start with the cube inscribed in the given sphere with
diameter D. We then draw pentagons which have the edges
of the cube as diagonals in the manner shown in the figure.
If H, iV, M, be the middle points of the sides of the face
BF, and H, G, L, K the middle points of the sides of the
face BD, join NO, GK which are then parallel to BC, and
draw MH, HL bisecting them at right angles at P, Q.
Divide PN, PO, QH in extreme and mean ratio at R, S, T,
and let PR, PS, QT be the greater segments. Draw RU, PX,
S V at right angles to the plane BF, and TW at right angles to
u
/
M |
F
//
/J
£— ■* — s
/
K /
W
jV
z
^^^^
f
a
the plane BD, such that each of these perpendiculars = PR
or PS. Join UV, VG, GW, WB, BU. These determine one
of the pentagonal faces, and the others are drawn similarly.
It is then proved that each of the pentagons, as UVCWB,
is (1) equilateral, (2) in the same plane, (3) equiangular.
As regards the sides we see, e. g., that
BU 2 = BR 2 + RU 2 = BN 2 + NR 2 + RP 2
= PN 2 + NR 2 + RP 2 = 4RP 2 (by means of XIII. 4) = UV 2 ,
and so on.
THE ELEMENTS. BOOK XIII 419
Lastly, it is proved that the same sphere of diameter D
which circumscribes the cube also circumscribes the dodeca-
hedron. For example, if Z is the centre of the sphere,
ZU 2 = ZX 2 + XU 2 = XS 2 + PS 2 = 3PN 2 , (XIII. 4)
while ZB 2 = 3ZP 2 = 3PN 2 .
If a be the edge of the dodecahedron, c the edge of the cube,
a= 2RP = 2 — c
4
2^3 \/5-l
~~ ~3~ ' ~~4
Book XIII ends with Proposition 18, which arranges the
edges of the five regular solids inscribed in one and the same
sphere in order of magnitude, while an addendum proves that
no other regular solid figures except the five exist.
The so-called Books XIV, XV.
This is no doubt the place to speak of the continuations
of Book XIII which used to be known as Books XIV, XV of
the Elements, though they are not by Euclid. The former
is the work of Hypsicles, who probably lived in the second
half of the second century B.C., and who is otherwise known
as the reputed author of an astronomical tract 'AvcupopiKos
(De ascensionibus) still extant (the earliest extant Greek book
in which the division of the circle into 360 degrees appears),
besides other works, which have not survived, on the harmony
of the spheres and on polygonal numbers. The preface to
' Book XIV ' is interesting historically. It appears from
it that Apollonius wrote a tract on the comparison of the
dodecahedron and icosahedron inscribed in one and the same
sphere, i. e. on the ratio between them, and that there were two
editions of this work, the first of which was in some way
incorrect, while the second gave a correct proof of the pro-
position that, as the surface of the dodecahedron is to
the surface of the icosahedron, so is the solid content of the
Ee 2
420 EUCLID
dodecahedron to that of the icosahedron, ' because the per-
pendicular from the centre of the sphere to the pentagon of
the dodecahedron and to the triangle of the icosahedron is the
same '. Hypsicles says also that Aristaeus, in a work entitled
Comparison of the five figures, proved that 'the same circle
circumscribes both the pentagon of the dodecahedron and the
triangle of the icosahedron inscribed in the same sphere ' ;
whether this Aristaeus is the same as the Aristaeus of the
Solid Loci, the elder contemporary of Euclid, we do not
know. The proposition of Aristaeus is proved by Hypsicles
as Proposition 2 of his book. The following is a summary
of the results obtained by Hypsicles. In a lemma at the end
he proves that, if two straight lines be cut in extreme and
mean ratio, the segments of both are in one and the same
ratio; the ratio is in fact 2:(\/5— 1). If then a ny straight
line A B be divided at G in extreme and mean ratio, AG being
the greater segment, Hypsicles proves that, if we have a cube,
a dodecahedron and an icosahedron all inscribed in the same
sphere, then :
(Prop. 7) (side of cube) : (side of icosahedron)
= -/(AW + AC*): V(AB 2 + BC 2 );
(Prop. 6) (surface of dodecahedron) : (surface of icosahedron)
= (side of cube) : (side of icosahedron) ;
(Prop. 8) (content of dodecahedron) : (content of icosahedron)
= (surface of dodecahedron) : (surface of icosahedron) ;
and consequently
(content of dodecahedron) : (content of icosahedron)
= V(AB 2 + AG 2 ): V(AB* + BC 2 ).
The second of the two supplementary Books (' Book XV ') is
also concerned with the regular solids, but is much inferior to
the first. The exposition leaves much to be desired, being
in some places obscure, in others actually inaccurate. The
Book is in three parts unequal in length. The first 1 shows
how to inscribe certain of the regular solids in certain others,
1 Heiberg's Euclid, vol. v, pp. 40-8.
THE] SO-CALLED BOOKS XIV, XV 421
(a) a tetrahedron in a cube, (b) an octahedron in a tetrahedron,
(c) an octahedron in a cube, (d) a cube in an octahedron,
(e) a dodecahedron in an icosahedron. The second portion 1
explains how to calculate the number of edges and the number
of solid angles in the five solids respectively. The third
portion 2 shows how to determine the dihedral angles between
the faces meeting in any edge of any one of the solids. The
method is to construct an isosceles triangle with vertical angle
equal to the said angle ; from the middle point of any edge
two perpendiculars are drawn to it, one in each of the two
faces intersecting in that edge ; these perpendiculars (forming
the dihedral angle) are used to determine the two equal sides
of an isosceles triangle, and the base of the triangle is easily
found from the known properties of the particular solid. The
rules for drawing the respective isosceles triangles are first
given all together in general terms ; and the special interest
of the passage consists in the fact that the rules are attributed
to ' Isidorus our great teacher '. This Isidorus is doubtless
Isidorus of Miletus, the architect of the church of Saint Sophia
at Constantinople (about A.D. 532). Hence the third portion
of the Book at all events was written by a pupil of Isidorus
in the sixth century.
The Data.
Coming now to the other works of Euclid, we will begin
with those which have actually survived. Most closely con-
nected with the Elements as dealing with plane geometry, the
subject-matter of Books I-VI, is the Data, which is accessible
in the Heiberg-Menge edition of the Greek text, and also
in the translation annexed by Simson to his edition of the
Elements (although this translation is based on an inferior
text). The book was regarded as important enough to be
included in the Treasury of Analysis (roVoy avaXvouevos) as
known to Pappus, and Pappus gives a description of it ; the
description shows that there were differences between Pappus's
text and ours, for, though Propositions 1-62 correspond to the
description, as also do Propositions 87-94 relating to circles
at the end of the book, the intervening propositions do not
1 Heiberg's Euclid, vol. v, pp. 48-50. 2 lb., pp. 50-66.
422 EUCLID
exactly agree, the differences, however, affecting the distribu-
tion and numbering of the propositions rather than their
substance. The book begins with definitions of the senses
in which things are said to be given. Things such as areas,
straight lines, angles and ratios are said to be ' given in
"magnitude when we can make others equal to them ' (Defs.
1-2). Rectilineal figures are 'given in species' when their
angles are severally given as well as the ratios of the sides to
one another (Def. 3). Points, lines and angles are 'given
in position ' ' when they always occupy the same place ' : a not
very illuminating definition (4). A circle is given in position
and in magnitude when the centre is given in position and
the radius in magnitude (6) ; and so on. The object of the
proposition called a Datum is to prove that, if in a given figure
certain parts or relations are given, other parts or relations are
also given, in one or other of these senses.
It is clear that a systematic collection of Data such as
Euclid's would very much facilitate and shorten the procedure
in analysis ; this no doubt accounts for its inclusion in the
Treasury of Analysis. It is to be observed that this form of
proposition does not actually determine the thing or relation
which is shown to be given, but merely proves that it can be
determined when once the facts stated in the hypothesis
are known ; if the proposition stated that a certain thing is
so and so, e.g. that a certain straight line in the figure is of
a certain length, it would be a theorem ; if it directed us to
find the thing instead of proving that it is 'given', it would
be a problem ; hence many propositions of the form of the
Data could alternatively be stated in the form of theorems or
problems.
We should naturally expect much of the subject-matter of
the Elements to appear again in the Data under the different
aspect proper to that book ; and this, proves to be the case.
We have already mentioned the connexion of Eucl. II. 5, 6
with the solution of the mixed quadratic equations ax ± x 2 = b 2 .
The solution of these equations is equivalent to the solution of
the simultaneous equations
y + x = a -)
xy — b 2 ) '
and Euclid shows how to solve these equations in Propositions
THE DATA 423
84, 85 of the Data, which state that * If two straight lines
contain a given area in a given angle, and if the difference
(sum) of them be given, then shall each of them be given.'
The proofs depend directly upon those of Propositions 58, 59,
' If a given area be applied to a given straight line, falling
short (exceeding) by a figure given in species, the breadths
of the deficiency (excess) are given.' All the 'areas' are
parallelograms.
We will give the proof of Proposition 59 (the case of
' excess '). Let the given area AB
be applied to A G, exceeding by the
figure GB given in species. I say
that each of the sides HC, GE is
given.
Bisect DE in F, and construct
on EF the figure FG similar and
similarly situated to GB (VI. 18).
Therefore FG, GB are about the same diagonal (VI. 26).
Complete the figure.
Then FG, being similar to GB, is given in species, and,
since FE is given, FG is given in magnitude (Prop. 52).
But AB is given; therefore AB + FG, that is to say, KL, is
given in magnitude. But it is also given in species, being
similar to GB; therefore the sides of KL are given (Prop. 55).
Therefore KH is given, and, since KG = EF is also given,
the difference GH is given. And GH has a given ratio to HB ;
therefore HB is also given (Prop. 2).
Eucl. III. 35, 36 about the 'power' of a point with reference
to a circle have their equivalent in Data 91, 92 to the effect
that, given a circle and a point in the same plane, the rectangle
contained by the intercepts between this point and the points
in which respectively the circumference is cut by any straight
line passing through the point and meeting the circle is
also given.
A few more enunciations may be quoted. Proposition 8
(compound ratio) : Magnitudes which have given ratios to the
same magnitude have a given ratio to one another also.
Propositions 45, 46 (similar triangles) : If a triangle have one
angle given, and the ratio of the sum of the sides containing
that angle, or another angle, to the third side (in each case) be
424 EUCLID
given, the triangle is given] in species. Proposition 52: If a
(rectilineal) figure given in species be described on a straight
line given in magnitude, the figure is given in magnitude.
Proposition 66 : If a triangle have one angle given, the rect-
angle contained by the sides including the angle has to the
(area of the) triangle a given ratio. Proposition 80 : If a
triangle have one angle given, and if the rectangle contained
by the sides including the given angle have to the square on
the third side a given ratio, the triangle is given in species.
Proposition 93 is interesting : If in a circle given in magni-
tude a straight line be drawn cutting off a segment containing
a given angle, and if this angle be bisected (by a straight line
cutting the base of the segment and the circumference beyond
it), the sum of the sides including the given angle will have a
given ratio to the chord bisecting the angle, and the rectangle
contained by the sum of the said sides and the portion of the
bisector cut off (outside the segment) towards the circum-
ference will also be given.
Euclid's proof is as follows. In the circle ABC let the
chord BC cut off a segment containing a given angle BAC,
and let the angle be bisected by AE meeting BC in D.
Join BE. Then, since the circle is given in magnitude, and
BC cuts off a segment containing a given
angle, BC is given (Prop. 87).
Similarly BE is given ; therefore the
ratio BC.BE is given. (It is easy to
see that the ratio BC : BE is equal to
2 cos \A)
Now, since the angle BAC is bisected,
BA : AC = BD : DC
It follows that (BA + AC) :(BD + DC)=AC: DC
But the triangles ABE, ADC are similar;
therefore AE : BE = AC : DC
= (BA + AC) : BC, from above.
Therefore (BA + AC) :AE = BC : BE, which is a given
ratio.
THE DATA 425
Again, since the triangles ADC, BDE are similar,
BE:ED = AC: CD = (BA + AC) : BC.
Therefore (BA +AC).ED = BC . BE, which is given.
On divisions (of figures).
The only other work' of Euclid in pure geometry which has
survived (but not in Greek) is the book On divisions (of
figures), nepl Siaipeo-ecov Pl^Xlov. It is mentioned by Proclus,
who gives some hints as to its content l ; he speaks of the
business of the author being divisions of figures, circles or
rectilineal figures, and remarks that the parts may be like
in definition or notion, or unlike ; thus to divide a triangle
into triangles is to divide it into like figures, whereas to
divide it into a triangle and a quadrilateral is to divide it into
unlike figures. These hints enable us to check to some extent
the genuineness of the books dealing with divisions of figures
which have come down through the Arabic. It was John Dee
who first brought to light a treatise De divislonibus by one
Muhammad Bagdadinus (died 1141) and handed over a copy
of it (in Latin) to Commandinus in 1563 ; it was published by
the latter in Dee's name and his own in 1570. Dee appears
not to have translated the book from the Arabic himself, but
to have made a copy for Commandinus from a manuscript of
a Latin translation which he himself possessed at one time but
which was apparently stolen and probably destroyed some
twenty years after the copy was made. The copy does not
seem to have been made from the Cotton MS. which passed to
the British Museum after it had been almost destroyed by
a fire in 1731. 2 The Latin translation may have been that
made by Gherard of Cremona (1114-87), since in the list of
his numerous translations a ' liber divisionum ' occurs. But
the Arabic original cannot have been a direct translation from
Euclid, and probably was not even a direct adaptation of it,
since it contains mistakes and unmathematical expressions ;
moreover, as it does not contain the propositions about the
1 Proclus on Eucl. I, p. 144. 22-6.
2 The question is fully discussed by R. C. Archibald, Euclid's Book on
Divisions of Figures with a restoration based on Woepcke's text and on the
Practica Geometriae of Leonardo Pisano (Cambridge 1915).
426 EUCLID
division of a circle alluded to by Proclus, it can scarcely have
contained more than a fragment of Euclid's original work.
But Woepcke found in a manuscript at Paris a treatise in
Arabic on the division of figures, which he translated and
published in 1851. It is expressly attributed to Euclid in the
manuscript and corresponds to the indications of the content
given by Proclus. Here we find divisions of different recti-
linear figures into figures of the same kind, e.g. of triangles
into triangles or trapezia into trapezia, and also divisions into
c unlike ' figures, e. g. that of a triangle by a straight line parallel
to the base. The missing propositions about the division of
a circle are also here : ' to divide into two equal parts a given
figure bounded by an arc of a circle and two straight lines
including a given angle ' (28), and ' to draw in a given circle
two parallel straight lines cutting off a certain fraction from
the circle ' (29). Unfortunately the proofs are given of only
four propositions out of 36, namely Propositions 19, 20, 28, 29,
the Arabic translator having found the rest too easy and
omitted them. But the genuineness of the treatise edited by
Woepcke is attested by the facts that the four proofs which
remain are elegant and depend on propositions in the
Elements, and that there is a lemma with a true Greek ring,
' to apply to a straight line a rectangle equal to the rectangle
contained by AB, AG and deficient by a square' (1 8). Moreover,
the treatise is no fragment, but ends with the words, ' end of
the treatise ', and is (but for the missing proofs) a well-ordered
and compact whole. Hence we may safely conclude that
Woepcke's tract represents not only Euclid's work but the
whole of it. The portion of the Practice/, geometriae of
Leonardo of Pisa which deals with the division of figures
seems to be a restoration and extension of Euclid's work ;
Leonardo must presumably have come across a version of it
from the Arabic.
The type of problem which Euclid's treatise was designed
to solve may be stated in general terms as that of dividing a
given figure by one or more straight lines into parts having
prescribed ratios to one another or to other given areas. The
figures divided are the triangle, the parallelogram, the trape-
zium, the quadrilateral, a figure bounded by an arc of a circle
and two straight lines, and the circle. The figures are divided
ON DIVISIONS OF FIGURES 427
into two equal parts, or two parts in a given ratio ; or again,
a given fraction of the figure is to be cut off, or the figure is
to be divided into several parts in given ratios. The dividing
straight lines may be transversals drawn through a point
situated at a vertex of the figure, or a point on any side, on one
of two parallel sides, in the interior of the figure, outside the
figure, and so on ; or again, they may be merely parallel lines,
or lines parallel to a base. The treatise also includes auxiliary
propositions, (1) ' to apply to a given straight line a rectangle
equal to a given area and deficient by a square ', the proposi-
tion already mentioned, which is equivalent to the algebraical
solution of the equation ax — x 2 = b 2 and depends on Eucl. II. 5
(cf. p. 152 above) ; (2) propositions in proportion involving
unequal instead of equal ratios :
If a . d > or < b . c, then a :b> or < c : d respectively.
If a : b > c:d, then (a + b):b > (c + d):d.
If a : b < c :d, then (a — b) : b < (c — d): d.
By way of illustration I will set out shortly three proposi-
tions from the Woepcke text.
(1) Propositions 19, 20 (slightly generalized): To cut off
a certain fraction (m/n) from a given triangle by a straight
line drawn through a given point within the triangle (Euclid
gives two cases corresponding to m/n =-§ and m/n = -§).
The construction will be best understood if we work out
the analysis of the problem (not given by Euclid).
Suppose that ABC is the given triangle, D the given
428 EUCLID
internal point ; and suppose the problem solved, i. e. GH
drawn through B in such a way that A GBH = — A ABC.
n
m
Therefore GB . BH = — . AB .BG. (This is assumed by
Euclid.)
Now suppose that the unknown quantity is GB — x, say.
Draw BE parallel to BC; then BE, EB are given.
Now BH: BE=GB:GE=x: (x-BE),
x.DE
or BH =
therefore GB.BH = x 2
x-BE'
DE
'x-BE'
And, by hypothesis, GB .BH = — . AB . BG;
n
therefore x z = — . — =— — (x — BE),
n BE v
.„ 7 m AB.BG , , A1
or, 11 A; = ^— — , we have to solve the equation
x 2 = k(x-BE),
or Jex-x 2 = k.BE.
This is exactly what Euclid does ; he first finds F on BA
such that BF.BE= m -AB.BC (the length of #F is deter-
mined by applying to BE a rectangle equal to —. AB .BC,
lb
Eucl. I. 45), that is, he finds BF equal to Jc. Then he gives
the geometrical solution of the equation kx — x 2 = k. BE in the
form ' apply to the straight line BF a rectangle equal to
BF .BE and deficient by a square'; that is to say, he deter-
mines G so that BG.GF= BF .BE. We have then only
to join GB and produce it to H; and GH cuts off the required
triangle.
(The problem is subject to a Siopio-fxos which Euclid does
not give, but which is easily supplied.)
(2) Proposition 28 : To divide into two equal parts a given
ON DIVISIONS OF FIGURES
429
figure bounded by an arc of a circle and by two straight lines
which form a given angle.
Let A BEG be the given figure, B the middle point of BG,
and BE perpendicular to BG. Join AB.
Then the broken line ABE clearly divides the figure into
two equal parts. Join AE, and draw
DF parallel to it meeting BA in F.
Join FE.
The triangles AFE, ABE are then
equal, being in the same parallels.
Add to each the area AEG.
Therefore the area AFEG is equal to the area ABEG, and
therefore to half the area of the given figure.
(3) Proposition 29 : To draw in a given circle two parallel
chords cutting off a certain fraction (m/ri) of the circle.
(The fraction m/n must be
such that we can, by plane
methods, draw a chord cutting off
m/n of the circumference of
the circle ; Euclid takes the case
where m/n = §.)
Suppose that the arc ABB is
m/n of the circumference of the
circle. Join A, B to the centre 0.
Draw OC parallel to A B and join
A C, BG. From B, the middle point
of the arc AB, draw the chord BE parallel to BG. Then shall
BG, BE cut off m/n of the area of the circle.
Since AB, OG are parallel,
AAOB = AACB.
Add to each the segment ABB ;
therefore
(sector ABBO) = figure bounded by AG, CB and arc ABB
= (segmt. ABC)-(segmt BFC).
Since BG, BE are parallel, (arc BB) = (arc GE) ;
430 EUCLID
therefore
(arc ABC) = (arc DCE), and (segmt. ABC) = (segmt. DCE) ;
therefore (sector AD BO), or — (circle ABC)
IV m
= (segmt. DCE)- (segmt. BFC).
Tilt
That is BC, DE cut off an area equal to — (circle ABC).
Lost geometrical works.
(a) The Pseudaria.
The other purely geometrical works of Euclid are lost so far
as is known at present. One of these again belongs to the
domain of elementary geometry. This is the Pseudaria, or
' Book of Fallacies ', as it is called by Proclus, which is clearly
the same work as the ' Pseudographemata ' of Euclid men-
tioned by a commentator on Aristotle in terms which agree
with Proclus's description. 1 Proclus says of Euclid that,
' Inasmuch as many things, while appearing to rest on truth
and to follow from scientific principles, really tend to lead one
astray from the principles and deceive the more superficial
minds, he has handed down methods for the discriminative
understanding of these things as well, by the use of which
methods we shall be able to give beginners in this study
practice in the discovery of paralogisms, and to avoid being
ourselves misled. The treatise by which he puts this machinery
in our hands he entitled (the book) of Pseudaria, enumerating
in order their various kinds, exercising our intelligence in each
case by theorems of all sorts, setting the true side by side
with the false, and combining the refutation of error with
practical illustration. This book then is by way of cathartic
and exercise, while the Elements contain the irrefragable and
complete guide to the actual scientific investigation of the
subjects of geometry.' 2
The connexion of the book with the Elements and the refer-
ence to its usefulness for beginners show that it did not go
beyond the limits of elementary geometry.
1 Michael Ephesius, Comm. on AHst. Soph. El., fol. 25 v , p. 76. 23 Wallies.
2 Proclus on Eucl. I, p. 70. 1-18. Cf. a scholium to Plato's Theaetetus
191 b, which says that the fallacies did not arise through any importation
of sense-perception into the domain of non-sensibles*
LOST GEOMETRICAL WORKS 431
' We now come to the lost works belonging to higher
geometry. The most important was evidently
(/?) The Porisms.
Our only source of information about the nature and con-
tents of the Porisms is Pappus. In his general preface about
the books composing the Treasury of Analysis Pappus writes
as follows 1 (I put in square brackets the words bracketed by
Hultsch).
' After the Tangencies (of Apollonius) come, in three Books,
the Porisms of Euclid, a collection [in the view of many] most
ingeniously devised for the analysis of the more weighty
problems, [and] although nature presents an unlimited num-
ber of such porisms, [they have added nothing to what was
originally written by Euclid, except that some before my time
have shown their want of taste by adding to a few (of the
propositions) second proofs, each (proposition) admitting of
a definite number of demonstrations, as we have shown, and
Euclid having given one for each, namely that which is the
most lucid. These porisms embody a theory subtle, natural,
necessary, and of considerable generality, which is fascinating
to those who can see and produce results].
' Now all the varieties of porisms belong, neither to theorems
nor problems, but to a species occupying a sort of intermediate
position [so that their enunciations can be formed like those of
either theorems or problems], the result being that, of the great
number of geometers, some regarded them as of the class of
theorems, and others of problems, looking only to the form of
the proposition. But that the ancients knew better the differ-
ence between these three things is clear from the definitions.
For they said that a theorem is that which is proposed with a
view to the demonstration of the very thing proposed, a pro-
blem that which is thrown out with a view to the construction
of the very thing proposed, and a porism that which is pro-
posed with a view to the producing of the very thing proposed.
[But this definition of the porism was changed by the more
recent writers who could not produce everything, but used
these elements and proved only the fact that that which is
sought really exists, but did not produce it, and were accord-
ingly confuted by the definition and the whole doctrine. They
based their definition on an incidental characteristic, thus :
A porism is that which falls short of a locus-theorem in
1 Pappus, vii, pp. 648-60.
432 EUCLID
respect of its hypothesis. Of this kind of porisms loci are
a species, and they abound in the Treasury of Analysis ; but
this species has been collected, named, and handed down
separately from the porisms, because it is more widely diffused
than the other species] . . . But it has further become charac-
teristic of porisms that, owing to their complication, the enun-
ciations are put in a contracted form, much being by usage
left to be understood ; so that many geometers understand
them only in a partial way and are ignorant of the more
essential features of their content.
' [Now to comprehend a number of propositions in one
enunciation is by no means easy in these porisms, because
Euclid himself has not in fact given many of each species, but
chosen, for examples, one or a few out of a great multitude.
But at the beginning of the first book he has given some pro-
positions, to the number of ten, of one species, namely that
more fruitful species consisting of loci.] Consequently, finding
that these admitted of being comprehended in our enunciation,
we have set it out thus :
If, in a system of four straight lines which cut one
another two and two, three points on one straight line
be given, while the rest except one lie on different straight
lines given in position, the remaining point also will lie
on a straight line given in position.
' This has only been enunciated of four straight lines, of
which not more than two pass through the same point, but it
is not known (to most people) that it is true of any assigned
number of straight lines if enunciated thus :
If any number of straight lines cut one another, not
more than two (passing) through the same point, and all
the points (of intersection situated) on one of them be
given, and if each of those which are on another (of
them) lie on a straight line given in position —
or still more generally thus :
if any number of straight lines cut one another, not more
than two (passing) through the same point, and all the
points (of intersection situated) on one of them be given,
while of the other points of intersection in multitude
equal to a triangular number a number corresponding
to the side of this triangular number lie respectively on
straight lines given in position, provided that of these
latter points no three are at the angular points of a
triangle (sc. having for sides three of the given straight
THE P0BI8MS 433
lines) — each of the remaining points will lie on a straight
line given in position. 1
' It is probable that the writer of the Elements was not
unaware of this, but that he only set out the principle ; and
he seems, in the case of all the porisms, to have laid down the
principles and the seed only [of many important things],
the kinds of which should be
istinguished according to the
differences, not of their hypotheses, but of the results and
the things sought. [All the hypotheses are different from one
another because they are entirely special, but each of the
results and things sought, being one and the same, follow from
many different hypotheses.]
' We must then in the first book distinguish the following-
kinds of things sought :
' At the beginning of the book is this proposition :
I. If from two given points straight lines be drawn
meeting on a straight line given in 'position, and one cut
off from a straight line given in position (a segment
measured) to a given point on it, the other will also cut
off from another {straight line a segment) having to thr
first a given ratio.
' Following on this (we have to prove)
II. that such and such a point lies on a straight line
given in position ;
III. that the ratio of such and such a pair of straight
lines is given ' ;
&c. &c. (up to XXIX).
'The three books of the porisms contain 38 lemmas: of the
theorems themselves there are 171.'
Pappus further gives lemmas to the Porisms?
With Pappus's account of Porisms must be compared the
passages of Proclus on the same subject. Proclus distinguishes
1 Loria (Le scienze esatte nell'antica Grecia, pp. 256-7) gives the mean-
ing of this as follows, pointing out that Simson first discovered it : 'If
a complete ^-lateral be deformed so that its sides respectively turn about
n points on a straight line, and (n — 1) of its \n{n -1) vertices move on
as many straight lines, the other \{n — 1) (n — 2) of its vertices likewise
move on as many straight lines : but it is necessary that it should be
impossible to form with the (w — 1) vertices any triangle having for sides
the sides of the polygon.'
2 Pappus, vii, pp. 866-918 ; Euclid, ed. Heiberg-Menge, vol. viii,
pp. 243-74.
1523 F f
434 EUCLID
the two senses of the word wopta-fia. The first is that of
a corollary, where something appears as an incidental result
of a proposition, obtained without trouble or special seeking,
a sort of bonus which the investigation has presented us
with. 1 The other sense is that of Euclid's Porisms. In
this sense
- porism is the name given to things which are sought, but
need some finding and are neither pure bringing into existence
nor simple theoretic argument. For (to prove) that the angles
at the base of isosceles triangles are equal is matter of theoretic
argument, and it is with reference to things existing that such
knowledge is (obtained). But to bisect an angle, to construct
a triangle, to cut off, or to place — all these things demand the
making of something ; and to find the centre of a given circle,
or to find the greatest common measure of two given commen-
surable magnitudes, or the like, is in some sort intermediate
between theorems and problems. For in these cases there is
no bringing into existence of the things sought, but finding
of them ; nor is the procedure purely theoretic. For it is
necessary to bring what is sought into view and exhibit it
to the eye. Such are the porisms which Euclid wrote and
arranged in three books of Porisms.' 2
Proclus's definition thus agrees well enough with the first,
the ' older ', definition of Pappus. A porism occupies a place
between a theorem and a problem ; it deals with something
already existing, as a theorem does, but h&s to find it (e.g. the
centre of a circle), and, as a certain operation is therefore
necessary, it partakes to that extent of the nature of a problem,
which requires us to construct or produce something not
previously existing. Thus, besides III. 1 and X. 3, 4 of the
Elements mentioned by Proclus, the following propositions are
real porisms: III. 25, VI. 11-13, VII. 33, 34, 36, 39, VIII. 2, 4,
X. 10, XIII. 18. Similarly, in Archimedes's On the Sphere and
Cylinder, I. 2-6 might be called porisms.
The enunciation given by Pappus as comprehending ten of
Euclid's propositions may not reproduce the form of Euclid's
enunciations; but, comparing the result to be proved, that
certain points lie on straight lines given in position, with the .
class indicated by II above, where the question is of such and
such a point lying on a straight line given in position, and
1 Proclus on Eucl. I, pp. 212. 14 ; 301. 22. 2 lb., p. 301. 25 sq.
THE P0RT8M8 435
with other classes, e.g. (V) that such and such a line is given
in position, (VI) that such and such a line verges to a given point,
(XXVII) that there exists a given point such that straight
lines drawn from it to such and such (circles) will contain
a triangle given in species, we may conclude that a usual form
of a porism was ' to prove that it is possible to find a point
with such and such a property ' or ' a straight line on which
lie all the points satisfying given conditions ', and so on.
The above exhausts all the positive information which we
have about the nature of a porism and the contents of Euclid's
Porism s. It is obscure and leaves great scope for speculation
and controversy ; naturally, therefore, the problem of restoring
the Porism s has had a great fascination for distinguished
mathematicians ever since the revival of learning. But it has
proved beyond them all. Some contributions to a solution have,
it is true, been made, mainly by Simson and Chasles. The first
claim to have restored the Porisms seems to be that of Albert
Girarcl (about 1590-1633), who spoke (1626) of an early pub-
lication of his results, which, however, never saw the light.
The great Fermat (1601-65) gave his idea of a 'porism',
illustrating it by five examples which are very interesting in
themselves ] ; but he did not succeed in connecting them with
the description of Euclid's Porisms by Pappus, and, though he
expressed a hope of being able to produce a complete restoration
of the latter, his hope was not realized. It was left for Robert
Simson (1687-1768) to make the first decisive step towards the
solution of the problem. 2 He succeeded in explaining the mean-
ing of the actual porisms enunciated in such general terms by
Pappus. In his tract on Porisms he proves the first porism
given by Pappus in its ten different cases, which, according to
Pappus, Euclid distinguished (these propositions are of the
class connected with loci) ; after this he gives a number of
other propositions from Pappus, some auxiliary proposi-
tions, and some 29 * porisms ', some of which are meant to
illustrate the classes I, VI, XV, XXVII-XXIX distin-
guished by Pappus. Simson was able to evolve a definition
of a porism which is perhaps more easily understood in
Chasles's translation : ' Le porisme est une proposition dans
1 (Euvres de Fermat, ed. Tannery and Henry, I, p. 76-84.
2 Roberti Simson Opera quaedam reliqua, 1776, pp. 315-594.
Ff 2
436 EUCLID
laquelle on demande de demontrer qu'une chose on plusieurs
choses sont donnees, qui, ainsi que Tune quelconque d'une
infinite d'autres choses non donnees, mais dont chacune est
avec des choses donnees dans une meme relation, ont une
proprie'te' commune, de'crite dans la proposition.' We need
not follow Simson's English or Scottish successors, Lawson
(1777), Playfair (1794), W. Wallace (1798), Lord Brougham
(1798), in their further speculations, nor the controversies
between the Frenchmen, A. J. H. Vincent and P. Breton (de
Champ), nor the latter's claim to priority as against Chasles ;
the work of Chasles himself (Les trois livres des Povismes
d'Euclide retablis . . . Paris, 1860) alone needs to be men-
tioned. Chasles adopted the definition of a porism given by
Simson, but showed how it could be expressed in a different
form. ' Porisms are incomplete theorems which express
certain relations existing between things variable in accord-
ance with a common law, relations which are indicated in
the enunciation of the porism, but which need to be completed
by determining the magnitude or position of certain things
which are the consequences of the hypotheses and which
would be determined in the enunciation of a theorem properly
so called or a complete theorem.' Chasles succeeded in eluci-
dating the connexion between a porism and a locus as de-
scribed by Pappus, though he gave an inexact translation of
the actual words of Pappus : ' Ce qui comtitue le porisme est
re qui manque a Vhypothcse d'un theoreme local (en d'autres
termes, le porisme est infeYieur, par l'hypothese, au theoreme
local ; c'est a dire que quand quelques parties d'une proposi-
tion locale n'ont pas dans l'e'nonce la determination qui leur
est propre, cette proposition cesse d'etre regarded comme un
theoreme et devient un porisme) ' : here the words italicized
are not quite what Pappus said, viz. that 'a porism is that
which falls short of a locus-theorem in respect of its hypo-
thesis ', but the explanation in brackets is correct enough if
we substitute ' in respect of ' for ' par ' (' by '). The work of
Chasles is historically important because it was in the course
of his researches on this subject that he was led to the idea of
anharmonic ratios ; and he was probably right in thinking
that the Porisms were propositions belonging to the modern
theory of transversals and to projective geometry. But, as a
THE PORISMS 437
restoration of Euclid's work, Chasles's Porisms cannot be re-
garded as satisfactory. One consideration alone is, to my
mind, conclusive on this point. Chasles made ' porisms ' out
of Pappus's various lemmas to Euclid's porisms and com-
paratively easy deductions from those lemmas. Now we
have experience of Pappus's lemmas to books which still
survive, e.g. the Conies of Apollonius; and, to judge by these
instances, his lemmas stood in a most ancillary relation to
the propositions to which they relate, and do not in the
least compare with them in difficulty and importance. Hence
it is all but impossible to believe that the lemmas to the
porisms were themselves porisms such as were Euclid's own
porisms ; on the contrary, the analogy of Pappus's other sets
of lemmas makes it all but necessary to regard the lemmas in
question as merely supplying proofs of simple propositions
assumed by Euclid without proof in the course of the demon-
stration of the actual porisms. This being so, it appears that
the problem of the complete restoration of Euclid's three
Books still awaits a solution, or rather that it will never be
solved unless in the event of discovery of fresh documents.
At the same time the lemmas of Pappus to the Porisms
are by no means insignificant propositions in themselves,
and, if the usual relation of lemmas to substantive proposi-
tions holds, it follows that the Porisms was a distinctly
advanced work, perhaps the most important that Euclid ever
wrote ; its loss is therefore much to be deplored. Zeuthen
has an interesting remark a propos of the proposition which
Pappus quotes as the first proposition of Book I, ' If from two
given points straight lines be drawn meeting on a straight
line given in position, and one of them cut off from a straight
line given in position (a segment measured) towards a given
point on it, the other w^ill also cut off from another (straight
line a segment) bearing to the first a given ratio.' This pro-
position is also true if there be substituted for the first given
straight line a conic regarded as the 'locus with respect to
four lines ', and the proposition so extended can be used for
completing Apollonius's exposition of that locus. Zeuthen
suggests, on this ground, that the Porisms were in part by-
products of the theory of conies and in part auxiliary means
for the study of conies, and that Euclid called them by the
438 EUCLID
same name as that applied to corollaries because they were
corollaries with respect to conies. 1 This, however, is a pure
conjecture.
(y) The Conies.
Pappus says of this lost work : ' The four books of Euclid's
Conies were completed by Apollonius, who added four more
and gave us eight books of Conies.' 2 It is probable that
Euclid's work was already lost by Pappus's time, for he goes
on to speak of ' Aristaeus who wrote the still extant five books
of Solid Loci avp^xv tois kcovlkoTs, connected with, or supple-
mentary to, the conies'. 3 This latter work seems to have
been a treatise on conies regarded as loci ; for ' solid loci ' was
a term appropriated to conies, as distinct from ' plane loci ',
which were straight lines and circles. In another passage
Pappus (or an interpolator) speaks of the ' conies ' of Aristaeus
the ' elder ', 4 evidently referring to the same book. Euclid no
doubt wrote on the general theory of conies, as Apollonius did,
but only covered the ground of Apollonius's first three books,
since Apollonius says that no one before him had touched the
subject of Book IV (which, however, is not important). As in
the case of the Elements, Euclid would naturally collect and
rearrange, in a systematic exposition, all that had been dis-
covered up to date in the theory of conies. That Euclid's
treatise covered most of the essentials up to the last part of
Apollonius's Book III seems clear from the fact that Apol-
lonius only claims originality for some propositions connected
with the ' three- and four-line locus ', observing that Euclid
had not completely worked out the synthesis of the said locus,
which, indeed, was not possible without the propositions
referred to. Pappus (or an interpolator) 5 excuses Euclid on
the ground that he made no claim to go beyond the discoveries
of Aristaeus, but only wrote so much about the locus as was
possible with the aid of Aristaeus's conies. We may conclude
that Aristaeus's book preceded Euclid's, and that it was, at
least in point of originality, more important. When Archi-
medes refers to propositions in conies as having been proved
1 Zeuthen, Die Lehrevon den Kegelschnitten im Alter turn, 1886, pp. 168,
173-4.
2 Pappus, vii, p. 672. 18. s Cf. Pappus, vii, p. 636. 23.
4 lb. vii, p. 672. 12. ° lb. vii, pp. 676. 25-678. 6.
THE C0N1CS AND SURFACE-LOCI 439
in the 'elements of conies', lie clearly refers to these two
treatises, and the other propositions to which he refers as well
known and not needing proof were doubtless taken from the
same sources. Euclid still used the old names for the conic
sections (sections of a right-angled, acute-angled, and obtuse-
angled cone respectively), but he was aware that an ellipse
could be obtained by cutting (through) a cone in any manner
by a plane not parallel to the base, and also by cutting a
cylinder ; this is clear from a sentence in his Lhaenomena to
the effect that, 'If a cone or a cylinder be cut by a plane not
parallel to the base, this section is a section of an acute-angled
cone, which is like a shield (0ujoeo?).'
(8) The Surf ace-Loci (tottol irpos kin(j)avda).
Like the Data and the Porisms, this treatise in two Books
is mentioned by Pappus as belonging to the Treasury of
Analysis. What is meant by surface-loci, literally ' loci on a
surface ' is not entirely clear, but we are able to form a con-
jecture on the subject by means of remarks in Proclus and
Pappus. The former says (l) that a locus is ' a position of a
line or of a surface which has (throughout it) one and the
same property ', l and (2) that ' of locus-theorems some are
constructed on lines and others on surfaces ' 2 ; the effect of
these statements together seems to be that ' loci on lines ' are
loci which are lines, and ' loci on surfaces ' loci which are
surfaces. On the other hand, the possibility does not seem to
be excluded that loci on surfaces may be loci traced on sur-
faces ; for Pappus says in one place that the equivalent of the
quadratrix can be got geometrically ' by means of loci on
surfaces as follows ' 3 and then proceeds to use a spiral de-
scribed on a cylinder (the cylindrical helix), and it is consis-
tent with this that in another passage 4 (bracketed, however, by
Hultsch) ' linear ' loci are said to be exhibited (Szikvvvtcli) or
realized from loci on surfaces, for the quadratrix is a ' linear '
locus, i.e. a locus of an order higher than a plane locus
(a straight line or circle) and a ' solid ' locus (a conic). How-
ever this may be, Euclid's Surface-Loci probably included
1 Proclus on Eucl. I, p. 394. 17. 2 76., p. 394. 19.
3 Pappus, iv, p. 258. 20-25. 4 lb. vii. 662. 9.
440 EUCLID
such loci as were cones, cylinders and spheres. The two
lemmas given by Pappus lend some colour to this view. The
first of these 1 and the figure attached to it are unsatisfactory
as they stand, but Tannery indicated a possible restoration. 2
If this is right, it suggests that one of the loci contained all
the points on the elliptical parallel sections of a cylinder, and
was therefore an oblique circular cylinder. Other assump-
tions with regard to the conditions to which the lines in the
figure may be subject would suggest that other loci dealt with
were cones regarded as containing all points on particular
parallel elliptical sections of the cones. In the second lemma
Pappus states and gives a complete proof of the focus-and-
directrix property of a conic, viz. that the locus of a point
the distance of which from a given point is in a given ratio
to its distance from a fixed straight line is a conic section,
which is an ellipse, a parabola or a hyperbola according as the
given ratio is less than, equal to, or greater than unity? Two
conjectures are possible as to the application of this theorem in
Euclid's Surface-Loci, (a) It may have been used to prove that
the locus of a point the distance of which from a given straight
line is in a given ratio to its distance from a given plane
is a certain cone. Or (6) it may have been used to prove
that the locus of a point the distance of which from a given
point is in a given ratio to its distance from a given plane is
the surface formed by the revolution of a conic about its major
or conjugate axis. 4
We come now to Euclid's works under the head of
Applied mathematics.
(a) The Phaenoniena.
The book on sphaeric intended for use in astronomy and
entitled Phaenomena has already been noticed (pp. 349, 351-2).
It is extant in Greek and was included in Gregory's edition of
Euclid. The text of Gregory, however, represents the later
of two recensions which differ considerably (especially in
Propositions 9 to 16). The best manuscript of this later
recension (b) is the famous Vat. gr. 204 of the tenth century
1 Pappus, vii, p. 1004. 17 ; Euclid, ed. Heiberg-Menge, vol. viii, p. 274.
2 Tannery in Bulletin des sciences maihematiques, 2 e serie, VI, p. 149.
3 Pappus!" vii, pp. 1004. 23-1014 ; Euclid, vol. viii, pp. 275-81.
4 For further details, see The Works of Archimedes, pp. lxii-lxv.
THE PHAENOMENA AND OPTICS 441
while the best manuscript of the older and better version (a)
is the Viennese MS.Vind. gr. XXXI. 13 of the twelfth century.
A new text edited by Menge and taking account of both
recensions is now available in the last volume of the Heiberg-
Menge edition of Euclid. 1
(ft) Optics and Catoptrlca.
The Optics, a treatise included by Pappus in the collection of
works known as the Little Astronomy, survives in two forms.
One is the recension of Theon translated by Zambertus in
1505; the Greek text was first edited by Johannes Pena
(de la Pene) in 1557, and this form of the treatise was alone
included in the editions up to Gregory's. But Heiberg dis-
covered the earlier form in two manuscripts, one at Vienna
(Vind. gr. XXXI. 13) and one at Florence (Laurent. XXVIII. 3),
and both recensions are contained in vol. vii of the Heiberg-
Menge text of Euclid (Teubner, 1895). There is no reason to
doubt that the earlier recension is Euclid's own work ; the
style is much more like that of the Elements, and the proofs of
the propositions are more complete and clear. The later recen-
sion is further differentiated by a preface of some length, which
is said by a scholiast to be taken from the commentary or
elucidation by Theon. It would appear that the text of this
recension is Theon's, and that the preface was a reproduction
by a pupil of what was explained by Theon in lectures. It
cannot have been written much, if anything, later than Theon's
time, for it is quoted by Nemesius about A.D. 400. Only the
earlier and genuine version need concern us here. It is
a kind of elementary treatise on perspective, and it may have
been intended to forearm students of astronomy against
paradoxical theories such as those of the Epicureans, who
maintained that the heavenly bodies are of the size that they
look. It begins in the orthodox fashion with Definitions, the
first of which embodies the same idea of the process of vision
as we find in Plato, namely that it is due to rays proceeding
from our eyes and impinging upon the object, instead of
the other way about : ' the straight lines (rays) which issue
from the eye traverse the distances (or dimensions) of great
1 Eudidis Phaenomena et scrrpta Musica edidit Henricus Menge.
Fraymenta collegit et disposuit J. L. Ileiberg, Teubner, 1916.
442 EUCLID
magnitudes ' ; Dei*. 2 : ' The figure contained by the visual rays
is a cone which has its vertex in the eye, and its base at the
extremities of the objects seen ' ; Def. 3 : ' And those things
are seen on which the visual rays impinge, while those are
not seen on which they do not ' ; Def. 4 : ' Things seen under
a greater angle appear greater, and those under a lesser angle
less, while things seen under equal angles appear equal ' ;
Def. 7 : 'Things seen under more angles appear more distinctly.'
Euclid assumed that the visual rays are not 'continuous',
i.e. not absolutely close together, but are separated by a
certain distance, and hence he concluded, in Proposition 1,
that we can never really see the whole of any object, though
we seem to do so. Apart, however, from such inferences as
these from false hypotheses, there is much in the treatise that
is sound. Euclid has the essential truth that the rays are
straight ; and it makes no difference geometrically whether
they proceed from the eye or the object. Then, after pro-
positions explaining the differences in the apparent size of an
object according to its position relatively to the eye, he proves
that the apparent sizes of two equal and parallel objects are
not proportional to their distances from the eye (Prop. 8) ; in
this proposition he proves the equivalent of the fact that, if a,
ft are two angles and a < ft < \ n, then
tan a oc
ttmft < ft'
the equivalent of which, as well as of the corresponding
formula with sines, is assumed without proof by Aristarchus
a little later. From Proposition 6 can easily be deduced the
fundamental proposition in perspective that parallel lines
(regarded as equidistant throughout) appear to meet. There
are four simple propositions in heights and distances, e.g. to
find the height of an object (1) when the sun is shining
(Prop. 18), (2) when it is not (Prop. 19) : similar triangles are,
of course, used and the horizontal mirror appears in the second
case in the orthodox manner, with the assumption that the
angles of incidence and reflection of a ray are equal, 'as
is explained in the Catoptrica (or theory of mirrors) '. Pro-
positions 23-7 prove that, if an eye sees a sphere, it sees
less than half of the sphere, and the contour of what is seen
OPTICS 443
appears to be a circle ; if the eye approaches nearer to
the sphere the portion seen becomes less, though it appears
greater ; if we see the sphere with two eyes, we see a hemi-
sphere, or more than a hemisphere, or less than a hemisphere
according as the distance between the eyes is equal to, greater
than, or less than the diameter of the sphere : these pro-
positions are comparable with Aristarchus's Proposition 2
stating that, if a sphere be illuminated by a larger sphere,
the illuminated portion of the former will be greater
than a hemisphere. Similar propositions with regard to the
cylinder and cone follow (Props. 28-33). Next Euclid con-
siders the conditions for the apparent equality of different
diameters of a circle as seen from an eye occupying various
positions outside the plane of the circle (Props. 34-7); he
shows that all diameters will appear equal, or the circle will
really look like a circle, if the line joining the eye to the
centre is perpendicular to the plane of the circle, or, not being
perpendicular to that plane, is equal to the length of the
radius, but this will not otherwise be the case (35), so that (36)
a chariot wheel will sometimes appear circular, sometimes
awry, according to the position of the eye. Propositions
37 and 38 prove, the one that there is a locus^such that, if the
eye remains at one point of it, while a straight line moves so
that its extremities always lie on it, the line will always
appear of the same length in whatever position it is placed
(not being one in which either of the extremities coincides
with, or the extremities are on opposite sides of, the point
at which the eye is placed), the locus being, of course, a circle
in which the straight line is placed as a chord, when it
necessarily subtends the same angle at the circumference or at
the centre, and therefore at the eye, if placed at a point of the
circumference or at the centre ; the other proves the same thing
for the case where the line is fixed with its extremities on the
locus, while the eye moves upon it. The same idea underlies
several other propositions, e.g. Proposition 45, which proves
that a common point can be found from which unequal
magnitudes will appear equal. The unequal magnitudes are
straight lines BC, CD so placed that BCD is a straight line.
A segment greater than a semicircle is described on BC, and
a similar segment on CD. The segments will then intersect
444 EUCLID
at F, and the angles subtended by BC and CD at F are
equal. The rest of the treatise is of the same character, and
it need not be further described.
The Catoptrica. published by Heiberg in the same volume is
not by Euclid, but is a compilation made at a much later date,
possibly by Theon of Alexandria, from aucient works on the
subject and mainly no doubt from those of Archimedes and
Heron. Theon l himself quotes a Catoptrica by Archimedes,
and Olympiodorus 2 quotes Archimedes as having proved the
fact which appears as an axiom in the Catoptrica now in
question, namely that, if an object be placed just out of sight
at the bottom of a vessel, it will become visible over the edge
when water is poured in. It is not even certain that Euclid
wrote Catoptrica^ at all, since, if the treatise was Theon's,
Proclus may have assigned it to Euclid through inadvertence.
(y) Music.
Proclus attributes to Euclid a work on the Elements of
Music (at Kara /iovo-lktju (TToi)(€i(o<r€is) 3 ; so does Marinus. 1
As a matter of fact, two musical treatises attributed to Euclid
are still extant, the tSectio Canonis (KaTarofii] kclvqvos) and the
Tntroductio harmonica (Elcraycoyr) ap/ioviKrj). The latter,
however, is certainly not b}^ Euclid, but by Cleonides, a pupil
of Aristoxenus. The question remains, in what relation does
the Sectio Canonis stand to the 'Elements' mentioned by
Proclus and Marinus ? The lectio gives the Pythagorean
theory of music, but is altogether too partial and slight to
deserve the title ' Elements of Music '. Jan, the editor of the
Musici Graeci, thought that the tiectio was a sort of summary
account extracted from the ' Elements ' by Euclid himself,
which hardly seems likely ; he maintained that it is the
genuine work of Euclid on the grounds (1) that the style and
diction and the form of the propositions agree well with what
we find in Euclid's Elements, and (2) that Porphyry in his
commentary on Ptolemy's Harmonica thrice quotes Euclid as
the author of a Sectio Canonis. 3 The latest editor, Menge,
' Theon, Comm. on Ptolemy's Syntaxis, i, p. 10.
2 Comment, on Arist. Meteorolog. ii, p. 94, Ideler, p. 211. 18 Busse.
;: Proclus on Eucl. I, p. 69. 3.
1 Marinus, Comm. on the Data (Euclid, vol. vi, p. 254. 19).
5 Sec Wallis, Opera mathematica, vol. iii, 1699, pp. 267, 269, 272.
ON MUSIC 445
points out that the extract given by Porphyry shows some
differences from our text and contains some things quite
unworthy of Euclid ; hence he is inclined to think that the
work as we have it is not actually by Euclid, but was ex-
tracted by some other author of less ability from the genuine
' Elements of Music ' by Euclid.
(8) Works on mechanics attributed to Euclid.
The Arabian list of Euclid's works further includes among
those held to be genuine ' the book of the Heavy and Light '.
This is apparently the tract Be levi et p)onderoso included by
Hervagius in the Basel Latin translation of 1537 and by
Gregory in his edition. That it comes from the Greek is
made clear by the lettering of the figures : and this is con-
firmed by the fact that another, very slightly different, version
exists at Dresden (Cod. Dresdensis Db. 86), which is evidently
a version of an Arabic translation from the Greek, since the
lettering of the figures follows the order characteristic of such
Arabic translations, a, b, g, d, e, z, It, t. The tract consists of
nine definitions or axioms and five propositions. Among the
definitions are these : Bodies are equal, different, or greater in
size according as they occupy equal, different, or greater spaces
(1-3). Bodies are equal in power or in virtue which move
over equal distances in the same medium of air or water in
equal times (4), while the power or virtue is greater if the
motion takes less time, and less if it takes more (6). Bodies
are of the same hind if, being equal in size, they are also equal
in power when the medium is the same ; they are different in
kind when, being equal in size, they are not equal in power or
virtue (7, 8). Of bodies different in kind, that has more power
which is more dense (solidivs) (9). With these hypotheses, the
author attempts to prove (Props. 1, 3, 5) that, of bodies which
traverse unequal spaces in equal times, that which traverses
the greater space has the greater power and that, of bodies of
the same kind, the poiver is proportional to the size, and con-
versely, if the power is proportional to the size, the bodies are
of the same kind. We recognize in the potentia or virtus
the same thing as the Svvaui? and i<r\v9 of Aristotle. 1 The
1 Aristotle, Physics, Z. 5.
446 EUCLID
property assigned by the author to bodies of the same kind is
quite different from what we attribute to bodies of the same
specific gravity ; he purports to prove that bodies of the
same kind have r po%wr proportional to their size, and the effect
of this, combined with the definitions, is that they move at
speeds proportional to their volumes. Thus the tract is the
most precise statement that we possess of the principle of
Aristotle's dynamics, a principle which persisted until Bene-
detti (1530-90) and Galilei (1564-1642) proved its falsity.
There are yet other fragments on mechanics associated with
the name of Euclid. One is a tract translated by Woepcke
from the Arabic in 1851 under the title ' Le livre d'Euclide
sur la balance ', a work which, although spoiled by some com-
mentator, seems to go back to a Greek original and to have
been an attempt to establish a theory of the lever, not from a
general principle of dynamics like that of Aristotle, but from
a few simple axioms such as the experience of daily life might
suggest. The original work may have been earlier than
Archimedes and may have been written by a contemporary of
Euclid. A third fragment, unearthed by Duhem from manu-
scripts in the Bibliotheque Nationale in Paris, contains four
propositions purporting to be 'liber Euclidis de ponderibus
secundum terminorum circumferentiam '. The first of the
propositions, connecting the law of the lever with the size of
the circles described by its ends, recalls the similar demon-
stration in the Aristotelian Mechanica ; the others attempt to
give a theory of the balance, taking account of the weight of
the lever itself, and assuming that a portion of it (regarded as
cylindrical) may be supposed to be detached and replaced by
an equal weight suspended from its middle point. The three
fragments supplement each other in a curious way, and it is a
question whether they belonged to one treatise or were due to
different authors. In any case there seems to be no indepen-
dent evidence that Euclid was the author of any of the
fragments, or that he wrote on mechanics at all. 1
1 For further details about these mechanical fragments see P. Duhem,
Les origines de la statique, 1905. esp. vol. i, pp. 61-97.
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